A348271 a(n) is the sum of noninfinitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset

1,4

Links

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

Eric Weisstein's World of Mathematics, Infinitary Divisor.

Formula

a(n) = A000203(n) - A049417(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^2, where c = A072691 - A327574 = 0.091748791296728992397... . - Amiram Eldar, Nov 25 2025

Example

a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1, n] - isigma[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n), d); sigma(f) - prod(i = 1, #f~, d = binary(f[i, 2]); prod(k = 1, #d, if(d[k], 1 + f[i, 1]^(2^(#d-k)), 1))); } \\ Amiram Eldar, Nov 25 2025

Crossrefs

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Cf. A072691, A327574.

Similar sequences: A034448, A048146, A051377, A188999.

Sequence in context: A092197 A213618 A319072 * A083804 A341840 A028597

Adjacent sequences: A348268 A348269 A348270 * A348272 A348273 A348274

Keyword

nonn

Author

Amiram Eldar, Oct 09 2021

Status

approved