

A281188


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


3



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
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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. AdamsWatters, 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^(n1), 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

Cf. A000005, A033950, A039819.
Sequence in context: A077618 A085863 A220354 * A323439 A054008 A125676
Adjacent sequences: A281185 A281186 A281187 * A281189 A281190 A281191


KEYWORD

nonn,easy


AUTHOR

Franklin T. AdamsWatters, Jan 16 2017


EXTENSIONS

More terms from Altug Alkan, Jan 17 2017


STATUS

approved



