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A281188
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Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.
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10
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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
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1,6
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = if(n==4, 1, if(issquarefree(n) == 1, omega(n)!, 0)); \\ Altug Alkan, Jan 18 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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