A281188 Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.

1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 6, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 6, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 2, 6, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 0, 2, 1, 0, 2, 2, 2, 0, 1, 0, 2, 0, 2, 2, 2, 0, 1, 0, 0, 0

Offset: 1

Author

Franklin T. Adams-Watters, Jan 16 2017

Keywords

• nonn
• easy

Comments

An integer n is a refactorable number if and only if tau(n) (A000005) divides n.

Every number is tau(m) for some refactorable m.

If n is squarefree with k prime divisors, then a(n) = k! (for a proof, see the Links entry from the author).

Conjecture: a(n) is nonzero if and only if n is squarefree or n = 4. [This conjecture is true; see Links for a proof. - Jon E. Schoenfield and Altug Alkan, Jan 17 2017]

See also Theorem 5 for the proof of conjecture in Colton link. - Altug Alkan, Jan 20 2017

Examples

			If n is prime, the only refactorable number m with tau(m) = n is n^(n-1), so a(n) = 1 for n prime.
Any number n of the form 8p, p a prime not equal to 2, has tau(n) = 8, and thus n is refactorable. Hence a(8) = 0.
		

Links

• Altug Alkan, Table of n, a(n) for n = 1..10000
• Franklin T. Adams-Watters, Refactorable numbers with tau squarefree
• S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
• Jon E. Schoenfield and Altug Alkan, Refactorable numbers with tau nonsquarefree

Crossrefs

Cf. A000005, A033950, A039819.

Programs

Mathematica

k = 1; t[] = 0; t[4] = 1; While[k < 100000001, m = DivisorSigma[0, k]; If[ Mod[k, m] == 0 && SquareFreeQ@ m, t[m]++]; k++]; t@# & /@ Range@20 (* _Robert G. Wilson v, Jan 16 2017 *)

PARI

a(n) = if(n==4, 1, if(issquarefree(n) == 1, omega(n)!, 0)); \\ Altug Alkan, Jan 18 2017

Extensions

More terms from Altug Alkan, Jan 17 2017