OFFSET
1,6
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
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
EXAMPLE
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.
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 *)
PROG
(PARI) a(n) = if(n==4, 1, if(issquarefree(n) == 1, omega(n)!, 0)); \\ Altug Alkan, Jan 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Franklin T. Adams-Watters, Jan 16 2017
EXTENSIONS
More terms from Altug Alkan, Jan 17 2017
STATUS
approved