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A281188
Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.
10
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
1,6
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
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
EXTENSIONS
More terms from Altug Alkan, Jan 17 2017
STATUS
approved