A348029 a(n) = A003959(n) - sigma(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors.

0, 0, 0, 2, 0, 0, 0, 12, 3, 0, 0, 8, 0, 0, 0, 50, 0, 9, 0, 12, 0, 0, 0, 48, 5, 0, 24, 16, 0, 0, 0, 180, 0, 0, 0, 53, 0, 0, 0, 72, 0, 0, 0, 24, 18, 0, 0, 200, 7, 15, 0, 28, 0, 72, 0, 96, 0, 0, 0, 48, 0, 0, 24, 602, 0, 0, 0, 36, 0, 0, 0, 237, 0, 0, 20, 40, 0, 0, 0, 300, 135, 0, 0, 64, 0, 0, 0, 144, 0, 54, 0, 48, 0

Offset: 1

Author

Antti Karttunen, Oct 20 2021

Keywords

• nonn

Comments

Inverse Möbius transform of A348030.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000203, A003959, A005117 (positions of zeros), A013661, A065488, A348030.

Programs

Mathematica

f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - DivisorSigma[1, n]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348029(n) = (A003959(n)-sigma(n));

Formula

a(n) = A003959(n) - A000203(n).

a(n) = Sum_{d|n} A348030(d).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2-p-1)) - Pi^2/6 = A065488 - A013661 = 1.0291786... . - Amiram Eldar, May 29 2025