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

Author

Altug Alkan, Jan 22 2017

Keywords

• nonn

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, ....

Examples

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

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

Crossrefs

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

Programs

PARI

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