A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

0, 0, 0, 4, 0, 0, 0, 18, 6, 0, 0, 16, 0, 0, 0, 64, 0, 18, 0, 24, 0, 0, 0, 72, 10, 0, 36, 32, 0, 0, 0, 210, 0, 0, 0, 94, 0, 0, 0, 108, 0, 0, 0, 48, 36, 0, 0, 256, 14, 30, 0, 56, 0, 108, 0, 144, 0, 0, 0, 96, 0, 0, 48, 664, 0, 0, 0, 72, 0, 0, 0, 342, 0, 0, 40, 80, 0, 0, 0, 384, 174, 0, 0, 128, 0, 0, 0, 216, 0, 108

Offset: 1

Author

Antti Karttunen, Oct 31 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21125

Crossrefs

Cf. A003959, A005117 (positions of zeros), A034448, A034460, A048146, A348029, A348507.

Programs

Mathematica

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348732(n) = (A003959(n)-A034448(n));

Formula

a(n) = A003959(n) - A034448(n).

a(n) = A348507(n) - A034460(n).

a(n) = A048146(n) + A348029(n).