Problem:
Score : 500 points

Problem Statement

Consider sequences \{A_1,...,A_N\} of length N consisting of integers between 1 and K (inclusive).

There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.

Since this sum can be enormous, print the value modulo (10^9+7).

Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.

Constraints

2 \leq N \leq 10^5

1 \leq K \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).

Sample Input 1

3 2

Sample Output 1

9

\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9

Thus, the answer is 9.

Sample Input 2

3 200

Sample Output 2

10813692

Sample Input 3

100000 100000

Sample Output 3

742202979

Be sure to print the sum modulo (10^9+7).

Solution in Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

module module_sort
    implicit none
    private
    public int_heap_sort,real_heap_sort
    interface sort
        module procedure int_heap_sort, real_heap_sort
    end interface

    contains

        subroutine int_heap_down(a,node,las)
            implicit none
            integer(8) a(*),x
            integer(8) node,par,ch1,ch2,las
            par = node
            x = a(par); ch1 = par*2
            do while(ch1 <= las)
                ch2 = ch1+1
                if(ch2 <= las .and. a(ch1) < a(ch2)) ch1 = ch2
                if(a(ch1) > x) then
                    a(par) = a(ch1); a(ch1) = x
                else 
                    exit
                end if
                par = ch1; ch1 = par*2
            end do
        end subroutine int_heap_down

        subroutine int_heap_sort(a,n)
            implicit none
            integer(8) a(*),x
            integer(8) n,i
            do i = n/2, 1, -1
                call int_heap_down(a,i,n)
            end do
            do i=n,2,-1
                x = a(i); a(i) = a(1); a(1) = x
                call int_heap_down(a,1_8,i-1)
            end do
        end subroutine int_heap_sort

        subroutine real_heap_down(a,node,las)
            implicit none
            real a(*), x
            integer(8) node,par,ch1,ch2,las
            par = node
            x = a(par); ch1 = par*2
            do while(ch1 <= las)
                ch2 = ch1+1
                if(ch2 <= las .and. a(ch1) < a(ch2)) ch1 = ch2
                if(a(ch1) > x) then
                    a(par) = a(ch1); a(ch1) = x
                else 
                    exit
                end if
                par = ch1; ch1 = par*2
            end do
        end subroutine real_heap_down

        subroutine real_heap_sort(a,n)
            implicit none
            real a(*),x
            integer(8) n,i
            do i = n/2, 1, -1
                call real_heap_down(a,i,n)
            end do
            do i=n,2,-1
                x = a(i); a(i) = a(1); a(1) = x
                call real_heap_down(a,1_8,i-1)
            end do
        end subroutine real_heap_sort

end module module_sort

module module_deque
    implicit none
    private
    type deque_node
        type(deque_node) ,pointer:: prev => null(),next => null()
        integer(8) val
    end type deque_node

    type,public:: deque
        type(deque_node) ,pointer:: first => null(),last => null(), ptr => null()
        integer(8):: size=0
        contains
            procedure:: push_back => deque_push_back
            procedure:: push_front => deque_push_front
            procedure:: pop_back => deque_pop_back
            procedure:: pop_front => deque_pop_front
    end type deque

    contains

        subroutine deque_push_back(self,val)
            class(deque)::self
            integer(8)::val
            if(self%size == 0) then
                allocate(self%first)
                self%last => self%first
                self%first%val = val
                self%size = 1
            else
                allocate(self%last%next)
                self%last%next%prev => self%last
                self%last => self%last%next
                self%last%val = val
                self%size = self%size+1
            end if
        end subroutine deque_push_back

        subroutine deque_push_front(self,val)
            class(deque)::self
            integer(8)::val
            if(self%size == 0) then
                allocate(self%first)
                self%last => self%first
                self%first%val = val
                self%size = 1
            else
                allocate(self%first%prev)
                self%first%prev%next => self%first
                self%first => self%first%prev
                self%first%val = val
                self%size = self%size+1
            end if
        end subroutine deque_push_front

        subroutine deque_pop_back(self)
            class(deque)::self
            if(self%size == 0) then
                return
            else if(self%size == 1) then
                self%size = 0
                deallocate(self%last)
                self%last => null(); self%first => null()
            else
                self%last => self%last%prev
                deallocate(self%last%next)
                self%last%next => null()
                self%size = self%size-1
            end if
        end subroutine deque_pop_back

        subroutine deque_pop_front(self)
            class(deque)::self
            if(self%size == 0) then
                return
            else if(self%size == 1) then
                self%size = 0
                deallocate(self%first)
                self%last => null(); self%first => null()
            else
                self%first => self%first%next
                deallocate(self%first%prev)
                self%first%prev => null()
                self%size = self%size-1
            end if
        end subroutine deque_pop_front

end module module_deque

module module_RedBlackTree
    implicit none
    private
    integer(8),parameter :: red = 1,black = 0

    type RBT_Int_node
        type(RBT_Int_node),pointer :: par=>null(), left => null(),right => null()
        integer(8)         :: key
        integer(8)         :: val=0
        integer(8)         :: color=red
    end type RBT_Int_node

    type RBT_Char_node
        type(RBT_Char_node),pointer :: par=>null(), left => null(),right => null()
        character(100)     :: key
        integer(8)         :: val=0
        integer(8)         :: color=red
    end type RBT_Char_node

    type(RBT_Int_node) ,pointer,save  :: rbt_int_nil
    type(RBT_Char_node) ,pointer,save :: rbt_char_nil

    type,public:: RedBlackTree_Int
        type(RBT_Int_node),pointer :: root => null()
        integer(8)                 :: size = 0
        contains
            procedure:: insert  => Int_Insert
            procedure:: get_val => Int_Get
            procedure:: find    => Int_Find
            procedure:: add     => Int_Add
            procedure:: FixUp   => Int_FixUp
    end type RedBlackTree_Int

    type,public:: RedBlackTree_Char
        type(RBT_Char_node),pointer :: root => null()
        integer(8)                  :: size = 0
        contains
            procedure:: insert  => Char_Insert
            procedure:: get_val => Char_Get
            procedure:: find    => Char_Find
            procedure:: add     => Char_Add
            procedure:: FixUp   => Char_FixUp
    end type RedBlackTree_Char

    contains

        subroutine Int_Init()
            implicit none

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if
        
        end subroutine  Int_Init

        recursive function Int_Get(self,key) result(val)
            implicit none
            class(RedBlackTree_Int), intent(in) :: self
            integer(8)                      :: key
            integer(8)                      :: val
            type(RBT_Int_node),pointer          :: u

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if

            if(.not.associated(self%root)) then
                val = -1 
                return
            end if

            u   => Int_SearchTree(self%root,key)
            val = u%val
            return

        end function Int_Get

        recursive function Int_Find(self,key) result(res)
            implicit none
            class(RedBlackTree_Int), intent(in) :: self
            integer(8)                          :: key
            logical                             :: res
            type(RBT_Int_node),pointer          :: u

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if

            if(.not.associated(self%root)) then
                res = .false.
                return
            end if

            u   => Int_SearchTree(self%root,key)
            res = (.not.associated(u,rbt_int_nil))
            return

        end function Int_Find

        recursive subroutine Int_Add(self,key,val)
            implicit none
            class(RedBlackTree_Int), intent(inout) :: self
            integer(8)                         :: key
            type(RBT_Int_node),pointer             :: u
            integer(8)                         :: val

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if

            if(.not.associated(self%root)) then
                call self%insert(key,val)
                return
            end if

            u   => Int_SearchTree(self%root,key)
            if (associated(u,rbt_int_nil)) then
                call self%insert(key,val)
            else
                u%val = u%val+val 
            end if
            return

        end subroutine Int_Add

        recursive function Int_SearchTree(u,key) result(res)
            implicit none
            type(RBT_Int_node), pointer :: u,res
            integer(8)              :: key

            if(associated(u,rbt_int_nil)) then
                res => rbt_int_nil
                return
            end if

            if(key < u%key) then
                res => Int_SearchTree(u%left,key)
                return
            else if(key > u%key) then
                res => Int_SearchTree(u%right,key)
                return
            else
                res => u
                return
            end if

        end function Int_SearchTree

        subroutine Int_Insert(self,key,val)
            implicit none
            class(RedBlackTree_Int),intent(inout):: self
            integer(8),intent(in)            :: key
            integer(8),intent(in)            :: val
            type(RBT_Int_node),pointer,save      :: now, u

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if

            !allocate new RBT_Int_node
            allocate(u)
            u%key = key; u%val = val
            u%left => rbt_int_nil; u%right => rbt_int_nil

            !insert new RBT_Int_node
            if(.not. associated(self%root)) then
                u%left => rbt_int_nil; u%right => rbt_int_nil; u%par => rbt_int_nil
                self%root => u
                self%root%color = black
                self%size = 1
                return
            else
                now => self%root
                if(.not.Insert_RBT_Int_node(u,now)) return
            end if

            !Fix Tree
            call self%FixUp(u)

            self%size = self%size+1
            
        end subroutine Int_Insert

        recursive function Insert_RBT_Int_node(u,now) result(added) 
            implicit none
            type(RBT_Int_node), pointer:: u,now
            logical            :: added
            
            if(u%key < now%key) then
                if(associated(now%left,rbt_int_nil)) then
                    now%left => u
                    u%par => now
                    added = .true.
                else
                    now => now%left
                    added = Insert_RBT_Int_node(u,now)
                end if
                return
            else if(u%key > now%key) then
                if(associated(now%right,rbt_int_nil)) then
                    now%right => u
                    u%par => now
                    added = .true.
                else
                    now => now%right
                    added = Insert_RBT_Int_node(u,now)
                end if
                return
            else
                added = .false.
                return
            end if

        end function Insert_RBT_Int_node 

        subroutine Int_FixUp(self,u)
            implicit none
            class(RedBlackTree_Int),intent(inout)     :: self
            type(RBT_Int_node),pointer,intent(inout)  :: u
            type(RBT_Int_node),pointer                :: w,g

            if(.not.associated(rbt_int_nil)) then
                allocate(rbt_int_nil)
                rbt_int_nil%color = black
            end if

            nullify(w,g)
            do while(u%color == red)
                if(u%key == self%root%key) then
                    u%color = black
                    return
                end if
                w => u%par
                if(w%left%color == black) then
                    call Int_FripLeft(w,self%root)
                    u => w
                    w => u%par
                end if
                if(w%color == black) return
                g => w%par
                if(g%right%color == black) then
                    call Int_FripRight(g,self%root)
                    return
                else
                    call Int_PushBlack(g)
                    u => g
                end if
            end do
        
        end subroutine Int_FixUp

        subroutine Int_PushBlack(u)
            implicit none
            type(RBT_Int_node), pointer:: u

            u%color = red; u%left%color = black; u%right%color = black; 

        end subroutine Int_PushBlack

        subroutine Int_PullBlack(u)
            implicit none
            type(RBT_Int_node), pointer:: u

            u%color = black; u%left%color = red; u%right%color = red; 

        end subroutine Int_PullBlack

        subroutine Int_FripLeft(u,root)
            implicit none
            type(RBT_Int_node), pointer, intent(inout) :: root
            type(RBT_Int_node), pointer                :: u,w
            integer(8)                             :: tmp
            
            tmp = u%color; u%color = u%right%color; u%right%color = tmp

            w => u%right
            w%par => u%par
            if(.not.associated(u%par,rbt_int_nil)) then
                if(associated(w%par%left,u)) then
                     w%par%left=>w
                else 
                    w%par%right=>w
                end if
            end if
            u%right => w%left
            if(.not.associated(u%right,rbt_int_nil))u%right%par => u
            u%par => w
            w%left => u
            if(associated(u,root)) then
                root => w
                root%par => rbt_int_nil
            end if

        end subroutine Int_FripLeft

        subroutine Int_FripRight(u,root)
            implicit none
            type(RBT_Int_node), pointer,intent(inout):: root
            type(RBT_Int_node), pointer              :: u,w
            integer(8)                           :: tmp

            tmp = u%color; u%color = u%left%color; u%left%color = tmp

            w => u%left
            w%par => u%par
            if(.not.associated(u%par,rbt_int_nil)) then
                if(associated(w%par%left,u)) then
                    w%par%left=>w
                else 
                    w%par%right=>w
                end if
            end if
            u%left => w%right
            if(.not.associated(u%left,rbt_int_nil))u%left%par => u
            u%par => w
            w%right => u
            if(associated(u,root)) then
                root => w
                root%par => rbt_int_nil
            end if

        end subroutine Int_FripRight

        subroutine Char_Init()
            implicit none
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if
        
        end subroutine  Char_Init
        
        recursive function Char_Get(self,key) result(val)
            implicit none
            class(RedBlackTree_Char), intent(in) :: self
            Character(100)                       :: key
            integer(8)                           :: val
            type(RBT_Char_node),pointer          :: u
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if

            if(.not.associated(self%root)) then
                val = -1
                return
            end if
        
            u   => Char_SearchTree(self%root,key)
            val = u%val
            return
        
        end function Char_Get
        
        recursive function Char_Find(self,key) result(res)
            implicit none
            class(RedBlackTree_Char), intent(in) :: self
            Character(100)                      :: key
            logical                             :: res
            type(RBT_Char_node),pointer         :: u
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if

            if(.not.associated(self%root)) then
                res = .false.
                return
            end if
        
            u   => Char_SearchTree(self%root,key)
            res = (.not.associated(u,rbt_char_nil))
            return
        
        end function Char_Find
        
        recursive subroutine Char_Add(self,key,val)
            implicit none
            class(RedBlackTree_Char), intent(inout) :: self
            Character(100)                         :: key
            type(RBT_Char_node),pointer            :: u
            integer(8)                             :: val
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if
        
            if(.not.associated(self%root)) then
                call self%insert(key,val)
                return
            end if
        
            u   => Char_SearchTree(self%root,key)
            if (associated(u,rbt_char_nil)) then
                call self%insert(key,val)
            else
                u%val = u%val+val 
            end if
            return
        
        end subroutine Char_Add
        
        recursive function Char_SearchTree(u,key) result(res)
            implicit none
            type(RBT_Char_node), pointer :: u,res
            Character(100)               :: key
        
            if(associated(u,rbt_char_nil)) then
                res => rbt_char_nil
                return
            end if
        
            if(key < u%key) then
                res => Char_SearchTree(u%left,key)
                return
            else if(key > u%key) then
                res => Char_SearchTree(u%right,key)
                return
            else
                res => u
                return
            end if
        
        end function Char_SearchTree
        
        subroutine Char_Insert(self,key,val)
            implicit none
            class(RedBlackTree_Char),intent(inout) :: self
            Character(100),intent(in)             :: key
            integer(8),intent(in)                 :: val
            type(RBT_Char_node),pointer,save      :: now, u
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if
        
            !allocate new RBT_Char_node
            allocate(u)
            u%key = key; u%val = val
            u%left => rbt_char_nil; u%right => rbt_char_nil
        
            !insert new RBT_Char_node
            if(.not. associated(self%root)) then
                u%left => rbt_char_nil; u%right => rbt_char_nil; u%par => rbt_char_nil
                self%root => u
                self%root%color = black
                self%size = 1
                return
            else
                now => self%root
                if(.not.Insert_RBT_Char_node(u,now)) return
            end if
        
            !Fix Tree
            call self%FixUp(u)
        
            self%size = self%size+1
            
        end subroutine Char_Insert
        
        recursive function Insert_RBT_Char_node(u,now) result(added) 
            implicit none
            type(RBT_Char_node), pointer:: u,now
            logical            :: added
            
            if(u%key < now%key) then
                if(associated(now%left,rbt_char_nil)) then
                    now%left => u
                    u%par => now
                    added = .true.
                else
                    now => now%left
                    added = Insert_RBT_Char_node(u,now)
                end if
                return
            else if(u%key > now%key) then
                if(associated(now%right,rbt_char_nil)) then
                    now%right => u
                    u%par => now
                    added = .true.
                else
                    now => now%right
                    added = Insert_RBT_Char_node(u,now)
                end if
                return
            else
                added = .false.
                return
            end if
        
        end function Insert_RBT_Char_node 
        
        subroutine Char_FixUp(self,u)
            implicit none
            class(RedBlackTree_Char),intent(inout)      :: self
            type(RBT_Char_node),pointer,intent(inout)  :: u
            type(RBT_Char_node),pointer                :: w,g
        
            if(.not.associated(rbt_char_nil)) then
                allocate(rbt_char_nil)
                rbt_char_nil%color = black
            end if
        
            nullify(w,g)
            do while(u%color == red)
                if(u%key == self%root%key) then
                    u%color = black
                    return
                end if
                w => u%par
                if(w%left%color == black) then
                    call Char_FripLeft(w,self%root)
                    u => w
                    w => u%par
                end if
                if(w%color == black) return
                g => w%par
                if(g%right%color == black) then
                    call Char_FripRight(g,self%root)
                    return
                else
                    call Char_PushBlack(g)
                    u => g
                end if
            end do
        
        end subroutine Char_FixUp
        
        subroutine Char_PushBlack(u)
            implicit none
            type(RBT_Char_node), pointer:: u
        
            u%color = red; u%left%color = black; u%right%color = black; 
        
        end subroutine Char_PushBlack
        
        subroutine Char_PullBlack(u)
            implicit none
            type(RBT_Char_node), pointer:: u
        
            u%color = black; u%left%color = red; u%right%color = red; 
        
        end subroutine Char_PullBlack
        
        subroutine Char_FripLeft(u,root)
            implicit none
            type(RBT_Char_node), pointer, intent(inout) :: root
            type(RBT_Char_node), pointer                :: u,w
            integer(8)                             :: tmp
            
            tmp = u%color; u%color = u%right%color; u%right%color = tmp
        
            w => u%right
            w%par => u%par
            if(.not.associated(u%par,rbt_char_nil)) then
                if(associated(w%par%left,u)) then
                     w%par%left=>w
                else 
                    w%par%right=>w
                end if
            end if
            u%right => w%left
            if(.not.associated(u%right,rbt_char_nil))u%right%par => u
            u%par => w
            w%left => u
            if(associated(u,root)) then
                root => w
                root%par => rbt_char_nil
            end if
        
        end subroutine Char_FripLeft
        
        subroutine Char_FripRight(u,root)
            implicit none
            type(RBT_Char_node), pointer,intent(inout):: root
            type(RBT_Char_node), pointer              :: u,w
            integer(8)                                :: tmp
        
            tmp = u%color; u%color = u%left%color; u%left%color = tmp
        
            w => u%left
            w%par => u%par
            if(.not.associated(u%par,rbt_char_nil)) then
                if(associated(w%par%left,u)) then
                    w%par%left=>w
                else 
                    w%par%right=>w
                end if
            end if
            u%left => w%right
            if(.not.associated(u%left,rbt_char_nil))u%left%par => u
            u%par => w
            w%right => u
            if(associated(u,root)) then
                root => w
                root%par => rbt_char_nil
            end if
        
        end subroutine Char_FripRight

end module module_RedBlackTree

module module_MinHeap
    implicit none
    private
    type MH_node
        type(MH_node),pointer :: par=>null(), left => null(),right => null()
        integer(8)            :: key,val
        integer(8)            :: size
    end type MH_node
    type(MH_node) ,pointer,save :: heap_nil

    type,public:: MinHeap
        type(MH_node),pointer :: root,last
        contains
            procedure:: push => MinHeap_Insert_MH_node
            procedure:: pop  => MinHeap_Pop_MH_node
    end type MinHeap

    contains

        subroutine MinHeap_Up(root,u)
            implicit none
            type(MH_node),pointer  :: root,u
            integer(8)             :: tmp

            do while(.not.associated(root,u))
                if(u%key < u%par%key) then
                    tmp        = u%key
                    u%key      = u%par%key
                    u%par%key  = tmp
                    tmp        = u%val
                    u%val      = u%par%val
                    u%par%val  = tmp
                    u => u%par
                else
                    return
                end if
            end do
            return

        end subroutine MinHeap_Up

        subroutine MinHeap_Down(root)
            implicit none
            type(MH_node),pointer:: root,u
            integer(8)           :: tmp

            u => root
            do while(.not.associated(u%left,heap_nil))

                if(.not.associated(u%right,heap_nil)) then
                    if (u%right%key < u%left%key .and. u%right%key < u%key) then
                        tmp         = u%key
                        u%key       = u%right%key
                        u%right%key = tmp
                        tmp         = u%val
                        u%val       = u%right%val
                        u%right%val = tmp
                        u => u%right
                        cycle
                    end if
                end if
                if(u%left%key < u%key) then
                    tmp        = u%key
                    u%key      = u%left%key
                    u%left%key = tmp
                    tmp        = u%val
                    u%val      = u%left%val
                    u%left%val = tmp
                    u => u%left
                    cycle
                end if
                exit
            end do
            return

        end subroutine MinHeap_Down

        subroutine Update_Size(root,u,is_add)
            implicit none
            type(MH_node),pointer :: root,u
            logical               :: is_add

            do while(.not.associated(root,u))
                if(is_add) then
                    u%size = u%size+1
                else
                    u%size = u%size-1
                end if
                u => u%par
            end do

            if(is_add) then
                u%size = u%size+1
            else
                u%size = u%size-1
            end if

        end subroutine Update_size

        subroutine MinHeap_Insert_MH_node(self,key,val)
            implicit none
            class(MinHeap)        :: self
            type(MH_node),pointer :: u,now
            integer(8)            :: key,val

            if(.not.associated(heap_nil)) allocate(heap_nil)
            u => null()
            allocate(u)
            u%key  = key
            u%val  = val
            u%left => heap_nil; u%right => heap_nil
            u%size = 1

            if(.not.associated(self%root)) then
                self%root      => u
                self%root%par  => heap_nil
                return
            end if

            now => self%root
            do
                now%size = now%size+1
                if(.not.associated(now%left,heap_nil)) then
                    if(.not.associated(now%right,heap_nil)) then
                        if(now%right%size < now%left%size) then
                            now => now%right; cycle
                        else
                            now => now%left;  cycle
                        end if
                    else
                        now%right => u
                        u%par     => now
                        exit
                    end if
                else
                    now%left => u
                    u%par    => now
                    exit
                end if
            end do

            call MinHeap_Up(self%root,u)

        end subroutine MinHeap_Insert_MH_node

        subroutine MinHeap_Pop_MH_node(self)
            implicit none
            class(MinHeap)        :: self
            type(MH_node),pointer :: u,now
            integer(8)            :: tmp

            if(.not.associated(heap_nil)) allocate(heap_nil)
            if(.not.associated(self%root)) return

            now => self%root
            do
                now%size = now%size-1
                if(.not.associated(now%left,heap_nil)) then
                    if(.not.associated(now%right,heap_nil)) then
                        if(now%right%size >= now%left%size) then
                            now => now%right; cycle
                        else
                            now => now%left;  cycle
                        end if
                    else
                        u => now%left
                        tmp           = u%key
                        u%key         = self%root%key
                        self%root%key = tmp
                        tmp           = u%val
                        u%val         = self%root%val
                        self%root%val = tmp
                        now%left => heap_nil
                        exit
                    end if
                else if(associated(now,self%root)) then
                    deallocate(self%root)
                    return
                else
                    u => now
                    tmp           = u%key
                    u%key         = self%root%key
                    self%root%key = tmp
                    tmp           = u%val
                    u%val         = self%root%val
                    self%root%val = tmp
                    if(associated(now%par%left,now)) then
                        now%par%left => heap_nil
                    else
                        now%par%right => heap_nil
                    end if
                    exit
                end if
            end do

            deallocate(u)

            call MinHeap_Down(self%root)

        end subroutine MinHeap_Pop_MH_node

end module module_MinHeap

recursive function  gcd(a,b) result(res)
    implicit none
    integer(8) :: a,b,res

    if(a < b) then
        res = gcd(b,a)
        return
    end if

    if(mod(a,b) == 0) then
        res = b
        return
    else 
        res = gcd(b,mod(a,b))
        return
    end if

end function gcd

recursive function mod_pow(a,b,modulo) result(res)
    implicit none
    integer(8) :: a,b,modulo,res

    if(b == 0) then
        res = 1
        return
    end if

    if(mod(b,2) == 1) then
        res = mod(a*mod_pow(a,b-1,modulo),modulo)
    else 
        res = mod(mod_pow(a,b/2,modulo)**2,modulo)
    end if

    return

end function mod_pow

module global
    use module_sort
    use module_deque
    use module_RedBlackTree
    use module_MinHeap
    implicit none
    integer(8) :: N,K,memo(100005),exa(100005)
    integer(8) :: modulo = 1e9_8+7_8
end module global

program main
    use global
    implicit none
    integer(8) :: i,j,mod_pow,ans = 0
    
    read*, N, K 

    do i = 1,K 
        memo(i) = mod_pow(K/i,N,modulo)
    end do

    do i = K, 1, -1
        exa(i) = memo(i)
        do j = i*2, K, i
            exa(i) = mod(exa(i)+modulo-exa(j),modulo)
        end do
        ans = mod(ans+exa(i)*i,modulo)
    end do

    print '(i0)', ans
    
end program main