A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

1, 1, 4, 1, 6, 4, 8, 1, 12, 6, 12, 4, 14, 8, 24, 1, 18, 12, 20, 6, 32, 12, 24, 4, 30, 14, 36, 8, 30, 24, 32, 1, 48, 18, 48, 12, 38, 20, 56, 6, 42, 32, 44, 12, 72, 24, 48, 4, 56, 30, 72, 14, 54, 36, 72, 8, 80, 30, 60, 24, 62, 32, 96, 1, 84, 48, 68, 18, 96, 48, 72, 12, 74, 38, 120, 20, 96, 56, 80, 6, 108, 42, 84, 32, 108

Offset: 1

Author

Antti Karttunen, Aug 31 2021

Keywords

• nonn
• mult

Comments

Coincides with A000593 on A122132.

Links

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

Crossrefs

Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.

Cf. also A000593, A053575.

Programs

Mathematica

f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)

PARI

A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));

Formula

Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.

a(n) = A001615(A000265(n)).

a(n) = A206787(n) * A336651(n). - Antti Karttunen, Feb 11 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Nov 19 2022

Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023