A318519 a(n) = A000005(n) * A003557(n).

1, 2, 2, 6, 2, 4, 2, 16, 9, 4, 2, 12, 2, 4, 4, 40, 2, 18, 2, 12, 4, 4, 2, 32, 15, 4, 36, 12, 2, 8, 2, 96, 4, 4, 4, 54, 2, 4, 4, 32, 2, 8, 2, 12, 18, 4, 2, 80, 21, 30, 4, 12, 2, 72, 4, 32, 4, 4, 2, 24, 2, 4, 18, 224, 4, 8, 2, 12, 4, 8, 2, 144, 2, 4, 30, 12, 4, 8, 2, 80, 135, 4, 2, 24, 4, 4, 4, 32, 2, 36, 4, 12, 4, 4, 4, 192, 2, 42, 18, 90, 2, 8, 2, 32, 8

Offset: 1

Author

Antti Karttunen, Sep 16 2018

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000005, A003557, A062355, A173557.

Programs

Mathematica

f[p_, e_] := (e + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2023 *)

PARI

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A318519(n) = (numdiv(n)*A003557(n));

PARI

A318519(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i,2]+1)*(f[i,1]^(f[i,2]-1))); };

Formula

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

a(n) = A000005(n) * A003557(n).

a(n) = A062355(n) / A173557(n).

Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - 2/p^(s-1) + 2/p^s - 1/p^(2*s-1) + 1/p^(2*s-2)). - Amiram Eldar, Sep 14 2023