aarjaneiro/planetmath huggingface dataset:
pandigital number

pandigital number

Given a base

b

integer

n

=

∑

i

=

1

k

d

i

⁢

b

i

-

1

where

d

1

is the least

significant digit

and

d

k

is the most significant, and

k

≥

b

, if for each

-

1

<

m

<

b

there is at least one

d

x

=

m

among the digits of

n

, then

n

is a

pandigital number

in base

b

.

The smallest pandigital number in base

b

is

b

b

-

1

+

∑

d

=

2

b

-

1

d

⁢

b

(

b

-

1

)

-

d

,

while the largest (with only one instance of each digit) is

∑

d

=

1

b

-

1

d

⁢

b

d

.

There are infinitely many pandigital numbers with more than one instance of one or more digits.

If

b

is not prime, a pandigital number must have at least

b

+

1

digits to be prime. With

k

=

b

for the length of digits of a pandigital number

n

, it follows from the

divisibility rules

in that base that

(

b

-

1

)

|

n

.

Sometimes a number with at least one instance each of the digits 1 through

b

-

1

but no instances of 0 is called a

zeroless pandigital number

.

Title

pandigital number

Canonical name

PandigitalNumber

Date of creation

2013-03-22 16:04:28

Last modified on

2013-03-22 16:04:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Defines

zeroless pandigital number

Generated on Fri Feb 9 15:23:23 2018 by

LaTeXML
parasitic number

parasitic number

Given a base

b

integer

n

with

k

digits

d

1

,

…

,

d

k

(with

d

1

being the least

significant digit

), and an integer

1

<

m

<

b

, if it is the case that

m

⁢

(

∑

i

=

1

k

d

i

⁢

b

i

-

1

)

=

d

1

⁢

b

k

-

1

+

(

∑

i

=

2

k

d

i

⁢

b

i

-

2

)

,

then

n

is called an

m

-

parasitic number

or

m

-

left-transposable number

.

If one takes an

m

-parasitic number, concatenates copies of its digits in order whatever number of times one wants, the resulting new number will also be

m

-parasitic.

Title

parasitic number

Canonical name

ParasiticNumber

Date of creation

2013-03-22 16:22:17

Last modified on

2013-03-22 16:22:17

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Synonym

left-transposable integer

Generated on Fri Feb 9 15:23:24 2018 by

LaTeXML
permutable prime

permutable prime

Given the base

b

representation

of a

prime number

p

as

d

k

,

…

,

d

1

with

p

=

∑

i

=

1

k

d

i

⁢

b

i

-

1

,

if each possible

permutation

of the digits still represents a prime number in that base, then

p

is said to be a

permutable prime

. For example, in base 10, the prime 337 is a permutable prime since 373 and 733 are also prime. The known base 10 permutable primes are listed in A003459 of Sloane’s OEIS.

If we define

π

P

⁢

(

n

)

to count how many permutable primes there are below

n

, it is obvious that

π

P

⁢

(

b

-

1

)

=

π

⁢

(

b

-

1

)

, where

π

⁢

(

n

)

is the standard

prime counting function

.

When

2

|

b

, a search for permutable primes can safely exclude any primes whose base

b

representation includes digits that are individually even. In a trivial sense, all

repunit primes

are also permutable primes. This means that in binary, the only permutable primes are

repunits

(that is, the

Mersenne primes

). Richert proved in 1951 that in the range

991

<

p

<

10

175

the only base 10 permutable primes are repunit primes; it is conjectured that this is also true above that range.

References

1

H. E. Richert, ”On

permutable

primtall,”

Unsolved Norsk Matematiske Tiddskrift

,

33

(1951), 50 - 54.

Title

permutable prime

Canonical name

PermutablePrime

Date of creation

2013-03-22 16:13:42

Last modified on

2013-03-22 16:13:42

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

absolute prime

Generated on Fri Feb 9 15:23:26 2018 by

LaTeXML
polydivisible number

polydivisible number

Given a base

b

integer

n

with

k

digits

d

1

,

…

,

d

k

, consider

d

k

the least

significant digit

and

d

1

, to suit our purpose in this case. If for each

1

<

j

<

k

it is the case that

(

∑

i

=

1

j

d

i

b

k

-

j

-

i

)

|

j

,

then

n

is said to be a

polydivisible number

.

A reasonably good

estimate

of how many polydivisible numbers base

b

has is

∑

i

=

2

b

-

1

(

b

-

1

)

⁢

b

i

-

1

i

!

.

In any given base, there is only one polydivisible number that is also a

pandigital number

.

Title

polydivisible number

Canonical name

PolydivisibleNumber

Date of creation

2013-03-22 16:22:20

Last modified on

2013-03-22 16:22:20

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:28 2018 by

LaTeXML
proof that every positive integer has a Zeckendorf representation

proof that every positive integer has a Zeckendorf representation

Theorem. Every

positive integer

n

has a

Zeckendorf representation

as a sum of non-consecutive

Fibonacci numbers

F

i

.

If an

integer

n

=

F

k

, where

F

x

refers to a Fibonacci number, then

Z

k

=

1

and

Z

i

=

0

for all

0

<

i

<

k

, where

Z

is an array of binary digits and

k

is the index of the most

significant digit

.

Otherwise, we assign

j

=

n

-

F

i

where

F

i

is the largest Fibonacci number such that

F

i

<

n

. Then

Z

i

>

0

. It is obvious that

j

<

F

i

, because it can be safely assumed at this juncture that

F

i

-

2

and

F

i

-

1

are distinct, that

F

i

-

2

<

F

i

-

1

and therefore

F

i

<

2

⁢

F

i

-

1

. This proves that

Z

i

must be 1, but no more than that. If

j

is a Fibonacci number, we can stop now, otherwise, we must again subtract the largest possible Fibonacci number, decrement

i

accordingly and set the appropriate

Z

i

=

1

. The farthest this iterative process can go is to

i

=

1

, corresponding to

F

1

=

1

.

Furthermore, the 1s in

Z

must have 0s in between them because any two consecutive 1s, such as at, say,

Z

i

and

Z

i

-

1

would indicate a failure to recognize that

F

i

-

1

+

F

i

=

F

i

+

1

, and thus

Z

i

and

Z

i

-

1

can be set to zero in favor of setting

Z

i

+

1

to 1. (As a side note, no binary representation of a

Mersenne number

should be mistaken for a Zeckendorf representation; in other words, there are no

repunits

in Zeckendorf representations).

Title

proof that every positive integer has a Zeckendorf representation

Canonical name

ProofThatEveryPositiveIntegerHasAZeckendorfRepresentation

Date of creation

2013-03-22 16:36:43

Last modified on

2013-03-22 16:36:43

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 11A63

Classification

msc 11B39

Generated on Fri Feb 9 15:23:30 2018 by

LaTeXML
proof that number of sum-product numbers in any base is finite

proof that number of sum-product numbers in any base is finite

Let

b

be the base of numeration.

Suppose that an

integer

n

has

m

digits when expressed in base

b

(not counting leading zeros, of course). Then

n

≥

b

m

-

1

.

Since each digit is at most

b

-

1

, we have that the sum of the digits
is at most

m

⁢

(

b

-

1

)

and the product is at most

(

b

-

1

)

m

, hence the
sum of the digits of

n

times the product of the digits of

n

is at
most

m

⁢

(

b

-

1

)

m

+

1

.

If

n

is a

sum-product number

, then

n

equals the sum of its digits
times the product of its digits. In light of the

inequalities

of the
last two paragraphs, this implies that

m

⁢

(

b

-

1

)

m

+

1

≥

n

≥

b

m

-

1

, so

m

⁢

(

b

-

1

)

m

+

1

≥

b

m

-

1

. Dividing both sides, we
obtain

(

b

-

1

)

2

⁢

m

≥

(

b

/

(

b

-

1

)

)

m

-

1

. By the

growth of exponential
function

, there can only be a finite number of values of

m

for which
this is true. Hence, there is a finite limit to the number of digits
of

n

, so there can only be a finite number of sum-product numbers to
any given base

b

.

Title

proof that number of sum-product numbers in any base is finite

Canonical name

ProofThatNumberOfSumproductNumbersInAnyBaseIsFinite

Date of creation

2013-03-22 15:47:06

Last modified on

2013-03-22 15:47:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 11A63

Generated on Fri Feb 9 15:23:31 2018 by

LaTeXML
proof that the set of sum-product numbers in base 10 is finite

proof that the set of sum-product numbers in base 10 is finite

This was first proven by David Wilson, a major contributor to Sloane’s

Online Encyclopedia of Integer Sequences

.

First, Wilson proved that

10

m

-

1

≤

n

(where

m

is the number of digits of

n

) and that

∑

i

=

1

m

d

i

≤

9

⁢

m

and

∏

i

=

1

m

d

i

≤

9

m

. The only way to fulfill the inequality

10

m

-

1

≤

9

m

⁢

9

⁢

m

is for

m

≤

84

.

Thus, a base 10

sum-product number

can’t have more than 84 digits. From the first

10

84

integers

, we can discard all those integers with 0’s in their decimal representation. We can further eliminate those integers whose

product

of digits is not of the form

2

i

⁢

3

j

⁢

7

k

or

3

i

⁢

5

j

⁢

7

k

.

Having thus

reduced

the number of integers to consider, a brute force search by

computer

yields the

finite set

of sum-product numbers in base 10: 0, 1, 135 and 144.

Title

proof that the set of sum-product numbers in base 10 is finite

Canonical

name

ProofThatTheSetOfSumproductNumbersInBase10IsFinite

Date of creation

2013-03-22 15:46:58

Last modified on

2013-03-22 15:46:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 11A63

Synonym

Proof that the set of sum-product numbers in decimal is finite

Generated on Fri Feb 9 15:23:33 2018 by

LaTeXML
proof that the set of sum-product numbers in base 2 is finite

proof that the set of sum-product numbers in base 2 is finite

For any

positive integer

with 0’s in its binary

representation

, the equality

n

=

∑

i

=

1

m

d

i

⁢

∏

i

=

1

m

d

i

is false, or we can rewrite it as the inequality

n

>

0

. Obviously, for

n

=

0

the equality given above is true, thus 0 is the only

sum-product number

in binary that has 0’s in its binary representation.

This leaves the

Mersenne numbers

(numbers whose significant digits are all 1’s), an infinite set but just as manageable. The Mersenne numbers are usually expressed as

2

k

-

1

, but for our purposes we express them as

∑

i

=

0

k

-

1

2

i

, which shows that for

n

=

2

k

-

1

,

∑

i

=

1

m

d

i

=

k

. Regardless of how many digits a Mersenne number has, it is obvious that

∏

i

=

1

m

d

i

=

1

because of the

multiplicative identity

. Thus, for a Mersenne number, the

right hand side

of our equality above will be

k

. For any nonnegative

k

, the inequality

k

<

2

k

-

1

will always hold except of course for

k

=

1

.

This proves that 0 and 1 are the only sum-product numbers in binary.

Title

proof that the set of sum-product numbers in base 2 is finite

Canonical name

ProofThatTheSetOfSumproductNumbersInBase2IsFinite

Date of creation

2013-03-22 15:46:56

Last modified on

2013-03-22 15:46:56

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

6

Author

Mravinci (12996)

Entry type

Proof

Classification

msc 11A63

Synonym

Proof that the set of sum-product numbers in binary is finite

Generated on Fri Feb 9 15:23:35 2018 by

LaTeXML
repdigit

repdigit

Given base

b

, a number of the form

d

⁢

(

b

n

-

1

b

-

1

)

for

n

>

0

and

0

<

d

<

b

is written using using the digit

d

only,

n

times in that base and is therefore a

repdigit

. The term, short for ”repeated digit,” is credited to Beiler’s book

Recreations in the theory of numbers

, in chapter 11.

When

d

=

1

, the resulting repdigit is called a

repunit

. Only repunits can also be prime (and then they are rare). No other repdigit can be prime since it is obvious that it is a

multiple

of a repunit.

In a trivial way, all repdigits are

palindromic numbers

.

Title

repdigit

Canonical name

Repdigit

Date of creation

2013-03-22 16:20:14

Last modified on

2013-03-22 16:20:14

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:36 2018 by

LaTeXML
repunit

repunit

Given base

b

, a number of the form

b

n

-

1

b

-

1

for

n

>

0

is written using using only the digit 1 in that base and is therefore a

repunit

. The term, short for ”repeated unit,” is credited to Beiler’s book

Recreations in the theory of numbers

, in chapter 11.

Regardless of base, a

prime number

is a prime number, but if in a given base it is a repunit, then it is called a

repunit prime

in that base. In binary, the

Mersenne numbers

are repunits, therefore the

Mersenne primes

are repunit primes in binary. Repunit primes in base 10 appear to be fewer, with only seven known as of 2006, for

n

taking on the values 2, 19, 23, 317, 1031, 49081, 86453 (see Sloane’s OEIS, A004023, for updates).

In a trivial way, repunit primes are also

palindromic primes

and

permutable primes

.

Title

repunit

Canonical name

Repunit

Date of creation

2013-03-22 16:13:26

Last modified on

2013-03-22 16:13:26

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Defines

repunit prime

Generated on Fri Feb 9 15:23:45 2018 by

LaTeXML
scientific notation

scientific notation

Scientific notation

is a manner of expressing

real numbers

in base 10 which provides a more

compact

way of writing very large or very small numbers without needing too many zeroes. To put it algebraically, a real

x

is expressed as

b

×

10

a

, where

b

is a

rational

real number with a limited number of

decimal places

, and

a

is an

integer

.

For example, 474200000000000000000000000000000000000000001701 is written as

4.742

×

10

47

in scientific notation, while the

reciprocal

of that number is written as

2.10881

×

10

-

48

in scientific notation. The number multiplied by the power of 10 is called the

mantissa

. It is customary to choose the mantissa

b

to be in the range

0

≤

|

b

|

<

10

so as to enable easier comparison of values. For example, it is clear from looking at the

exponents

alone, that of the numbers

3.1403

×

10

97

and

-

4.58990321

×

10

1729

, the latter has the

greater

absolute value

.

It is understood that some loss of precision is acceptable for the application at hand; that it is not necessary to know the least significant digit, or even hundreds of digits besides the few most significant to be put in the mantissa. This would be unacceptable in most applications of

number theory

, but it is adequate and even necessary for many applications in scientific fields such as physics, biology, seismography, etc.

Most

scientific calculators

support scientific notation, with the 10 tacit and possibly the letter “E” (for exponent). Hardware

calculators

might be limited to exponents

-

100

<

a

<

100

. Software calculators are usually not subject to this limitation.

Title

scientific notation

Canonical

name

ScientificNotation

Date of creation

2013-03-22 16:39:13

Last modified on

2013-03-22 16:39:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:46 2018 by

LaTeXML
self number

self number

An

integer

n

that in a given base

b

lacks a

digitaddition generator

.

Consider, for example, the integer 41 in base 10. It can be expressed as 34 + 3 + 4. For 42, however, there is no such digitaddition, hence it is a self number.

If

2

|

b

, all odd

n

<

b

will be self numbers.

Though self numbers form a small proportion of most ranges of

2

⁢

b

consecutive integers, there are infinitely many of them: The

recurrence relation

S

i

=

(

b

-

2

)

⁢

b

i

-

1

+

S

i

-

1

+

(

b

-

2

)

(with

S

1

=

b

-

1

if

2

|

b

and

S

1

=

b

-

2

otherwise) will give an incomplete though

infinite

list of self numbers.

Reference

Kaprekar, D. R.

The Mathematics of New

Self-Numbers

. Devaiali, 1963: 19 - 20

Title

self number

Canonical

name

SelfNumber

Date of creation

2013-03-22 15:56:15

Last modified on

2013-03-22 15:56:15

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Synonym

Columbian number

Synonym

Colombian number

Synonym

self-number

Generated on Fri Feb 9 15:23:48 2018 by

LaTeXML
self-descriptive number

self-descriptive number

A

self-descriptive number

n

in base

b

is an

integer

such that each base

b

digit

d

x

=

∑

d

i

=

x

1

where each

d

i

is a digit of

n

,

i

is a very simple, standard iterator operating in the range

-

1

<

i

<

b

, and

x

is a position of a digit; thus

n

“describes” itself.

For example, the integer 6210001000 written in base 10. It has six instances of the digit 0, two instances of the digit 1, a single instance of the digit 2, a single instance of the digit 6 and no instances of any other base 10 digits.

Base 4 might be the only base with two self-descriptive numbers,

1210

4

and

2020

4

. From base 7 onwards, every base

b

has at least one self-descriptive number of the form

(

b

-

4

)

b

-

1

+

2

⁢

b

b

-

2

+

b

b

-

3

+

b

4

. It has been proven that 6210001000 is the only self-descriptive number in base 10, but it’s not known if any higher bases have any self-descriptive numbers of any other form.

Sequence

A108551 of the OEIS lists self-descriptive numbers from quartal to

hexadecimal

.

Title

self-descriptive number

Canonical name

SelfdescriptiveNumber

Date of creation

2013-03-22 15:53:27

Last modified on

2013-03-22 15:53:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

self descriptive number

Generated on Fri Feb 9 15:23:50 2018 by

LaTeXML
sequences b^{2⁢n}-1 and b^{2⁢n-1}+1 are divisible by b+1

sequences

b

2

⁢

n

-

1

and

b

2

⁢

n

-

1

+

1

are divisible by

b

+

1

Consider the

alternating

geometric finite series

S

m

+

1

⁢

(

μ

)

=

∑

i

=

0

m

(

-

1

)

i

+

μ

⁢

b

i

,

(1)

where

μ

=

1

,

2

and

b

≥

2

an

integer

. Multiplying (1) by

-

b

and subtracting from it

(

b

+

1

)

⁢

S

m

+

1

⁢

(

μ

)

=

∑

i

=

0

m

(

-

1

)

i

+

μ

⁢

b

i

-

∑

i

=

0

m

(

-

1

)

i

+

1

+

μ

⁢

b

i

+

1

and by elemental manipulations, we obtain

S

m

+

1

⁢

(

μ

)

=

(

-

1

)

μ

⁢

[

1

-

(

-

1

)

m

+

1

⁢

b

m

+

1

]

b

+

1

=

∑

i

=

0

m

(

-

1

)

i

+

μ

⁢

b

i

.

(2)

Let

μ

=

1

,

m

=

2

⁢

n

-

1

. Then

b

2

⁢

n

-

1

b

+

1

=

-

∑

i

=

0

2

⁢

n

-

1

(

-

1

)

i

⁢

b

i

.

(3)

Likewise, for

μ

=

2

,

m

=

2

⁢

n

-

2

b

2

⁢

n

-

1

+

1

b

+

1

=

∑

i

=

0

2

⁢

n

-

2

(

-

1

)

i

⁢

b

i

,

(4)

as desired.

Palindromic numbers of even length

As an application of above

sequences

, let us consider an even

palindromic number

(EPN) of arbitrary length

2

⁢

n

which can be expressed in any base

b

as

(

E

⁢

P

⁢

N

)

n

=

∑

k

=

0

n

-

1

b

k

⁢

(

b

2

⁢

n

-

1

-

k

+

b

k

)

=

∑

k

=

0

n

-

1

b

k

b

k

⁢

[

(

b

2

⁢

n

-

1

+

1

)

+

(

b

2

⁢

k

-

1

)

]

,

(5)

where

0

≤

b

k

≤

b

-

1

.It is clear, from (3) and (4), that

(

E

⁢

P

⁢

N

)

n

is

divisible

by

b

+

1

. Indeed this one can be given by

(

E

⁢

P

⁢

N

)

n

=

(

b

+

1

)

⁢

∑

k

=

0

n

-

1

∑

j

=

0

2

⁢

(

n

-

1

-

k

)

b

k

⁢

(

-

1

)

j

⁢

b

k

+

j

.

(6)

Title

sequences

b

2

⁢

n

-

1

and

b

2

⁢

n

-

1

+

1

are divisible by

b

+

1

Canonical name

SequencesB2n1AndB2n11AreDivisibleByB1

Date of creation

2013-03-22 16:14:19

Last modified on

2013-03-22 16:14:19

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

6

Author

perucho (2192)

Entry type

Derivation

Classification

msc 11A63

Generated on Fri Feb 9 15:23:53 2018 by

LaTeXML
Sloane’s conjecture on multiplicative digital root

Sloane’s conjecture on multiplicative digital root

It is believed that there is no

integer

has a multiplicative persistence greater than itself, a

conjecture

put forth in 1973 by Neil Sloane, and that Sloane meant to limit this conjecture to fixed radix bases.

In 1998,

Diamond

and Reidpath published a

factorial base

counterexample

, by proving that “it is possible to find a number in factorial base of arbitrarily large persistence,” specifically a number of the form

n

!

⁢

n

+

∑

i

=

1

n

-

1

i

!

Obviously, this number will have a factorial base multiplicative digital root of

n

and a persistence of also

n

, suggesting an

upper bound

for the desired counterexample.

References

1

M. R. Diamond, D. D. Reidpath, “A Counterexample to Conjectures by Sloane and Erdos Concerning the Persistence of Numbers”,

J. Rec. Math.

29 (1998), 89 - 92.

2

N. J. A. Sloane, “The persistence of a number”

J. Rec. Math.

6 (1973), 97 - 98.

Title

Sloane’s conjecture on multiplicative digital root

Canonical

name

SloanesConjectureOnMultiplicativeDigitalRoot

Date of creation

2013-03-22 16:00:45

Last modified on

2013-03-22 16:00:45

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Conjecture

Classification

msc 11A63

Synonym

Sloane-Erdős conjecture on multiplicative digital root

Generated on Fri Feb 9 15:23:55 2018 by

LaTeXML
Smarandache-Wellin number

Smarandache-Wellin number

Given a base

b

, concatenate the base

b

representations

of the first

n

primes into a single

integer

, placing the first prime as the most significant digit(s) and the

n

th prime as the least significant digit(s). This is the

Smarandache-Wellin number

S

n

.

For example, in base 10,

S

8

is 235711131719, the

concatenation

of the strings “2”, “3”, “5”, “7”, “11”, “13”, “17” and “19” reinterpreted as a single integer.

Placing a

decimal point

immediately preceding a base 10 Smarandache-Wellin number turns it into an approximation of the

Copeland-Erdos constant

.

References

R. Crandall and C. Pomerance,

Prime Numbers

: A Computational Perspective, Springer, NY, 2001: 72

H. Ibstedt, A Few

Smarandache Sequences

,

Smarandache Notions Journal

, Vol. 8, No. 1-2-3, 1997: 170 - 183

Title

Smarandache-Wellin number

Canonical

name

SmarandacheWellinNumber

Date of creation

2013-03-22 15:55:01

Last modified on

2013-03-22 15:55:01

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

9

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Related topic

FlorentinSmarandache

Generated on Fri Feb 9 15:23:56 2018 by

LaTeXML