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

1,12

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Infinitary Divisor.

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Similar sequences: A034444, A048105, A286324, A049419.

Sequence in context: A262889 A083059 A173440 * A330936 A284687 A046268

Adjacent sequences: A348338 A348339 A348340 * A348342 A348343 A348344

Keyword

nonn

Author

Amiram Eldar, Oct 13 2021

Status

approved