A348341 a(n) is the number of noninfinitary divisors of n.

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Oct 13 2021

Keywords

• nonn

Examples

			a(4) = 1 since 4 has one noninfinitary divisor, 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Infinitary Divisor.

Crossrefs

Cf. A000005, A036537, A037445, A077609, A162643, A327576, A348271.

Similar sequences: A034444, A048105, A286324, A049419.

Programs

Mathematica

a[1] = 0; a[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

PARI

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2]))); \\ Antti Karttunen, Oct 13 2021

Formula

a(n) = A000005(n) - A037445(n).

a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).

a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (1 - 2 * A327576) = 0.266749... . - Amiram Eldar, Dec 09 2022