A340368 Multiplicative with a(p^e) = (p - 1) * (p + 1)^(e-1).

1, 1, 2, 3, 4, 2, 6, 9, 8, 4, 10, 6, 12, 6, 8, 27, 16, 8, 18, 12, 12, 10, 22, 18, 24, 12, 32, 18, 28, 8, 30, 81, 20, 16, 24, 24, 36, 18, 24, 36, 40, 12, 42, 30, 32, 22, 46, 54, 48, 24, 32, 36, 52, 32, 40, 54, 36, 28, 58, 24, 60, 30, 48, 243, 48, 20, 66, 48, 44, 24, 70, 72, 72, 36, 48, 54, 60, 24, 78, 108, 128, 40

Offset: 1

Author

Antti Karttunen, Jan 06 2021

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383

Crossrefs

Cf. A167344, A173557, A327564, A340323.

Programs

Mathematica

f[p_, e_] := (p - 1)*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 12 2022 *)

PARI

A340368(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]-1)*((f[i,1]+1)^(f[i,2]-1))));

Formula

a(n) = A167344(n) / A340323(n).

a(n) = A173557(n) * A327564(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 3/p^2 + 2/p^4) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.4313799748... . - Amiram Eldar, Nov 12 2022