Problem:
Score : 600 points

Problem Statement

Given is a positive integer N.

We will choose an integer K between 2 and N (inclusive), then we will repeat the operation below until N becomes less than K.

Operation: if K divides N, replace N with N/K; otherwise, replace N with N-K.

In how many choices of K will N become 1 in the end?

Constraints

2 \leq N \leq 10^{12}

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of choices of K in which N becomes 1 in the end.

Sample Input 1

6

Sample Output 1

3

There are three choices of K in which N becomes 1 in the end: 2, 5, and 6.

In each of these choices, N will change as follows:

When K=2: 6 \to 3 \to 1

When K=5: 6 \to 1

When K=6: 6 \to 1

Sample Input 2

3141

Sample Output 2

13

Sample Input 3

314159265358

Sample Output 3

9

Solution in Rust (1.15.1) (judge verdict: Accepted)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use contest::scanner::Scanner;
use contest::prime_number::divisors;

fn main() {
    let mut scan = Scanner::new();
    let n = scan.read::<i64>();
    let c1 = divisors(n-1).len()-1;
    let d = divisors(n);
    let mut c2 = 0;
    for i in d {
        if i == 1 {continue;}
        let mut e = n;
        while e % i == 0 {
            e /= i;
        }
        if e % i == 1 {
            c2 += 1;
        }
    }
    println!("{}", c1 + c2);
}
*/

fn main() {
    let exe = "/tmp/binDECFF867";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &[u8]) -> Vec<u8> {
    let mut tbl = [0; 256];
    for i in 0..85 { tbl[TBL[i] as usize] = i as u32; }
    let mut v_vec = v.to_vec();
    let padding = (5 - v.len() % 5) % 5;
    v_vec.extend(vec![b'~'; padding]);
    let mut ret = vec![0u8;v_vec.len() / 5 * 4];
    for (i, ck) in v_vec.chunks(5).enumerate() {
        let mut acc = 0;
        for &c in ck {
            acc = acc * 85 + tbl[c as usize];
        }
        ret[i * 4..(i + 1) * 4].copy_from_slice(&unsafe{std::mem::transmute::<u32, [u8; 4]>(acc.to_be())});
    }
    let l = ret.len();
    ret.resize(l - padding, 0u8);
    ret
}

const TBL: &'static [u8] = b"0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz!#$%&()*+-;<=>?@^_`{|}~";
const BIN: &'static [u8] = b"e??420s#R700000000000suY$0RR910INX&00000KmY&$00000000000000000000Kma%Z0sue&000000RR911poj50000000000002M$00000002M$00000?6Ls>00000?6Ls>00000001BW000000RR911^@s6sGI`;00000sGMT}00000sGMT}0000000000000000000000000001BW00000w3t?$RZv(V^al+V00000@Pz{a@Pz{a$N>NVr~m)}0ssI2`}-mPe??420s#R40suY$4PXO64;%7r76kzCfdc>&fgQVhH~<F)5C9J=1pojoAy3#q2f$tf0Nd>Q&L99c1`THy2V=rMJ0Un4Er>M$H?jpAYy<<yvIjteOvn=?000CB!H%$C3vVzd^81#s$ZY^n<#c2>tQJC^kZlJ;ya@mbY<F01Q8xwv00;|q@(=)0Z>L}g3H#s}00+4M000000FVIx_*?@3<e~up0!a-3|KD`W1OOBR14m9(0CbXi98>TlUb+AO|NnwTBfJ6tu%?2o#86nn06|wlRx!XqRY6oqgUtU`Q%Sd%|Nn{pOo`7<jVuC4gX~C1i8LfkBnT7XOq=^}cLyl)t`1Zr7)XnhB!F2H_x_K(Z4bx|j3kIeBp^ul&`OOQAphIj-$5KJO^tg26I>(^1pep^0h6>yg}_?-HTQIWOp8DSDv8HBVDGth*9%xl0mTO@SxefseQO&JiP7_<1i=z?fVQ=5`w7kh?-$t$5ww1``))f(jX69;LdFCb0AXVELv9=F*@QlejG+1-?xt`5x9>rT#DetI#7IPm;9FfmTtNd~L0`?scK^dwOpOi3>;g;;j|LU62Lm;?hTppq4MDhY0gUBS4hR$l6nonW-DM+O7%~&zPy%jA2m8L;J#7>OQZ^(QiF_ms4E+T5XLs%XC<e12ixj1ONHOg(+1u^i=UtQhzyB`e@D%_7s$AQ<x7$|(i#Lr0-m!*@4S{=VMGN1x+cC)(J?9VJ0#S-)kgZd<ZQK1&fIk9r2*?Oqz{5@Z^mhmgxA*Nyi#`HI2>%cT8+Zuu3GTzbcN6~=6MV3HFaIS15VgYL{bnu?AA|q^|NB$j{)Vz5M{6nZNIB6DgOmdEo8$Ja4gvww4}uE=4i<+DxAx(0NIBU^4@cR@Ax&Ke+jN`TZiO*i$BEu(60Ro1H^bNcO@-uja_K++9YY=(hk9*z|0GxIXh_EdBzgzF2;FXO+XpeyAzeIFbxg`6B-mzi%_cWh6<6yeFZ&@U2Q3^EO^GI9Vn-NkUgccaz5z6>d@mWE38W7T2DJJ-WZSj}L<GYRQ2!T3HmohOSrrry*gi}VFD(xsao3}3`eY<BjcyZ01Af(gaKC*y_;m*zdk_M0K?^?wLZPwIuozR^OWutLXKlx}quz@=G!R7C4{rr{FZ^#8gZKe;I`csff4xb^2DwA}OK(F2k`Gs#4-rKGxdCk(<7|)K(&Zq)=}P|acHw50e#fNNLm-<zK!E?=*nZo6I1g~ZPG(6N#0fkkNxZ@M@ogVOBp?bFFxyrxOYwPB=LqO+N9^V**2hKb<co8fZ5ucDIsS502s8o4j^qx<Z8QA8k(I=B2Z{zkhzIKor~=|V26yReQ9MK00pREbTWA}O$SF^W5t?Sx8xAbO!nnhcJ|xl)V<C@7f$aF;9br2>1)BrJvoOxT4Q?--%>-<YGXD>R4+py4TPheNBlo~T2e1q>$RMDQfPJ2Ci%D<<U{FBEV?P;~Ix*m@R{?K0?S|<K(>0-H0)fblTn=ErF~+|OP2PP$|JJ_;412DJFMcV*$J53+a$9I^bPYBZ_NknAfv@}9_7@lebXC3p9ls#Dl{S<6@wp;Sxc>m<DT1N<9%j=BJ%SKW7&g})E*4omK3P`N%{5#_Ec*;(Jvj@6#v5V31Q#Ad02WyJv47OZ0BvDuZp#lS1*5I*+Z!9cLpXZU6u8F^5d7%()@%<KDF@JGfjlKBlK0r#SOY^-4~qmPf{lP67U<q~G!8q#0b|JzjvwV4;p_fL0uP0;1f)2{C`*uSEjDeRZBSKjn+(Wfm?=o8#vb9dAdjt0gF66N9$U9?SOX+gV|#mUbZAT4judAg@Czt12|?yeZ`ZHP)}%v_Ie)d*GPwjbHGCZQ>+!ZG1Fnr7RNITob_lk2;S8eVYhMPWFfY8@o04AiiNi+);46s(bz<GH&c=z@h{RKB(AcwgheTnS22<-I4~fGP?Aw-jsqjWGmI_f&m)<eBz`oGUUAb>*Hg@g}{0P2AQ5;t^{Ki4pFyX5tFgTaDW3c-HQdH}RG%nPOJaja^N3$c8XD`{L=)M6EhaB6P6e{S+U#V&snq#{q9(dD9x^pH;#&$~?->f*0<T;s;5c1I{mcWFxuT>D&LHLle9T0e8Uv(2dUS016C=Zq%<pC0(%hob%6)3=F4vz{aImX>2=@Z$%2*X-*65VY0Hf%;1qpxvL#~eA7fNZBr$i}}4C^1hhQpb+Vx*-Q2rEKg$avPUWzbID%E*oPR2JW&oKrIp0*NjbGKp?mhDM5I2abPBx3zZhL*t+>6hOBdg#=ds8OLhopQwra1GY2kZ=N1Ttb8=nGT8nXrg9a=)cK5+XGrvfTb#vdHO-(+<pdzY2SP;fQ7;%H&e19-AiA5QSP3MVyhxJ>a$(AVM4=CeBS(<NwFp)qhAR|G5FbID{MTGWrDqD_LBi$zC19K<>1&{$f90aquVFn+QSlDu{#kPPatZfH%5Su`(xQi}{K{{9`c*vp%0rjImVFu<M9x{4JcBul|jf6rH#+hb)6Qc)^1uQn)O{fIZF%uMuh&%$q126r63wrl#BvOJ<f^oUq9Za*t0tO`>#p@uk35AxBABseSrS{klwncPuL<wI}>?mw~bWIM+J{k}l{{LtE88AGJNI{KG_kj5Ya>7A?{vdR``yDmhAhyOvd7#2b!RiJ=4_X|$a5Oqm{stH38|*jQNEAj7qr!5N$N@XVR<NHM5E;>Mi-CsW@aBsNnZhpHx4%#)iXT9=0kSLmD|8D4E$?rIUoH<|Ay@eUbQgruiEh))(GRf%JOTL+4`wcWHw#ZS29QrcgzYjCHF#KWK#g<%UNmF`?Q6HT+F(NtSO8XbP%gJDZkZs4#{X<6gPW5vl1xO6G{@Q!je+!u`OZwYYrw$v&U8c}Ddlbbnr%A<Ztew>5rZt)kd``GgzYjjNSW~cciY<D71DI}C;IrncgF!kxY~4fb-Ycwl)6m;Qxaz*EZyMhga?VApPX%CaM~*lsNBs0K{ps9=xgu?2H-;nhnc_!0R|8QYX|4RL&z_5Ya(fPI<P*Jk8e%AS&<FK561_eSOTbE<ktn8H-a8;6?g$0zQJm|T692EW*BTu3!fBFE8T1;BNPmNK=kWimKlZH;b4UJzNMJ^4M%ku<N_H?j7;^xT>^0bP&l-TlWsecKY=bV3WPUX_Te);1O*Q5?=A93TPkGASkxeXVPk}B@}&-*C&s$5`G740?isDif^eiDNl1;chm*d@*%IOoh$D#|vS%ZA8>SEem=1?*vV&McCV(Cj13pM^b2A3p<PQsk%mW7ohz&bb%s!yp2O){VD7R-Zt35SgE09ACbt6c8-mp>ogS>CVx}D#2_KUpo#m;UcFh&83013Wb6V_X)bf{VSG24GRmT770(R6w*GJep%Gsk|*^$8D$#71k|3J-t@gTi<0TcZIOI`eVV=?}*PIdx@+y9*5Sx{gmO&IK^ne-}_qNlyohBil$RP1Z5(i3`WQEH-=(&9PBrpA79?Bk<Z44ddoqa^?@TWrC)N+qq;EI|O2}cDQvEH;cYj_=(sW4@8JG#fV5G@V0M>5w#;b_7JP<D>;Z9IIztY0;DtyT(x~mB&Q$4JRo<UxxfN3ZZ`k`x`x5yfPFJy$b{Q?CBurk*#maV4hX~h5QgM)AS1BB>C-t3|Ce~MOb@r%O5Q=i?goS7Kf&V$2-&d+J-Px>NB=`S0^N;K0u@4y6f=Rsz`@zSclKt96bNmXNVot1fTqF`Xt-bZ+0w4)5o+Sgb@h54fE71YSOZRIVK7EFqobzT3`TdO3hjdHu~1maz=&zm=05{rf`h`&fTHRM#kG_z83M27j_eLI>#AOkPpha@NOb{eIXX6;W3+p|2|g$-$;3ShrY7W*R3N@9GFXbm*4erS5;(34T#H{EGP&k$dKhbm<k_th2~WNwA6B%6T_ScSzmO**09}_Ii}#QL*1iZ%zYiffvM3Npe^oZYfCRuex{EH1--M=v2ieC5Oq%xyJaGe5RR{qT5=<R<{MNPvWE~<c2vPz7syJlv7wU!EbvNs5fB)4|wh<>2b>c_F65GQzbT<|ITtVLA8;y&Kg**>+Jrn|R#S%Dp+7Mi9J4viJHxj%BtEUrK>!9hkT@%F=Rq#YHbk6GukaM)rfvyn>!(q!1Km+X~ekei425*}o?S1<$1x5j5FzH4R>l%t9cf0Fa*)<0w06~Zz>jViPU|kCB#5mf9XhgE_Z~U__Y{82k84#<4X(Ip$=7I~Kx`TUlbP6;OwgxuLsiOu|Xzq3FFk%a6K68#MEG%L~uy4Z<;vZ{V#19K^^$19FWJ{?zQW1#3tA?LdXEy&Y#EqSc2R#mpZcvHoK=#E1bSy;Uw;czA$q0!{)DFsfHq~B`n_1YN0~*B)3T6bvc7Lf*GFfJ?X9Oe<jZiRA`1baVa92Dmw!%OI)(=9^85-cWD2L0y1MOe9P%GlWb<`Ap3?#pOVmn5GDbI%gmvb$n5Nq41RJ&lg9{u68n`;j_^m$o!iEGmkSU4Lxm{-0mulERcKBUi%($0fap%ltKcoy--L4dq07S^J67)`j)h-YFJ$pmU9Xj)K;rclQbWZ;H&j-^p##2_RAxVB%!`b~5JmUKr6RMy(AIdLc=z~l_TA;bvmD0KsLnm9FA`-oXB2m~DjTxBb;FDP{cU{`)|pa*|I<FAQ~>}&3<c_<VF9w;cp$Pr?cY$=Q>9hG34vzs%4fkSMt6XFmLD1Rs-C^C?EyB!fjaS=;!5mPvoHl1$QjTg=H5H%ImAPly(89@XY(G5j_961->7-x+&%!rB&&<B)qB|I4Nj3n?CzzZOQ4A9^<Fg+aIFGxoxMIi|dEs6&qAsKCitcD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