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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 (list; graph; refs; listen; history; text; internal format)
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

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. Adams-Watters, Jan 16 2017

EXTENSIONS

More terms from Altug Alkan, Jan 17 2017

STATUS

approved

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)