A389171 a(n) = n + A048146(n) - A057723(n), where A048146 is sum of non-unitary divisors and A057723 is sum of positive divisors of n that are divisible by every prime that divides n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 0, 2, 3, 0, 0, 14, 0, 5, 0, 2, 0, 12, 0, 6, 0, 0, 0, 18, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 5, 2, 0, 0, 0, 14, 0, 0, 0, 22, 0, 0, 0, 6, 0, 24, 0, 2, 0, 0, 0, 30, 0, 7, 3, 7, 0, 0, 0, 6

Offset

1,12

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = n - A389170(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + zeta(2) * (1 - 1/zeta(3) - Product_{p prime} (1 - 1/p^2 + 1/p^3)) = 0.0452101403394169979608... . - Amiram Eldar, Oct 05 2025

Mathematica

f1[p_, e_] := (p^(e+1) - 1)/(p-1);
f2[p_, e_] := (p^(e+1)-1)/(p-1) - 1;
a[n_] := Module[{f = FactorInteger[n]}, n + Times @@ f1 @@@ f - Times @@ (1 + Power @@@ f) - Times @@ f2 @@@ f]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 05 2025 *)

Prog

(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048146(n) = (sigma(n)-A034448(n));
A057723(n) = { my(f=factor(n)~); prod(i=1, #f, sigma(f[1, i]^f[2, i])-1); };
A389171(n) = ((n+A048146(n))-A057723(n));

Crossrefs

Cf. A005117, A034448, A048146, A057723, A389170.

Cf. A126706 (positions of positive terms), A303554 (of 0's).

Sequence in context: A284270 A337542 A282568 * A028833 A024943 A325787

Adjacent sequences: A389168 A389169 A389170 * A389172 A389173 A389174

Keyword

nonn,easy

Author

Antti Karttunen, Oct 04 2025

Status

approved