A281495 Least k > 1 such that k^n is a refactorable number.

2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 9, 15, 2, 17, 6, 19, 10, 21, 8, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 9, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78, 79

Offset

1,1

Comments

Theorem: There are infinitely many n-th power refactorable numbers for any given value of n > 1.

For proof see Alkan link.

Numbers n such that a(n) is not equal to A007947(n+1) are 13, 21, 40, 85, 121, 171, 182, 208, 312, 341, 364, 514, 562, 585, 661, 665, 781, ...

Primes p such that a(p-1) is not equal to p are 41, 313, 563, 1013, 1201, 1823, ....

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Altug Alkan, Proof of Theorem

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.

Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8

Example

a(4) = 5 because 625 = 5^4 is the least fourth power refactorable number that is greater than 1.

Prog

(PARI) isA033950(n) = n % numdiv(n) == 0;
a(n) = my(k=2); while (!isA033950 (k^n), k++); k;

Crossrefs

Cf. A001597, A007947, A033950, A036907, A235524.

Sequence in context: A165743 A086297 A261969 * A056554 A346486 A088835

Adjacent sequences: A281492 A281493 A281494 * A281496 A281497 A281498

Keyword

nonn

Author

Altug Alkan, Jan 22 2017

Status

approved