A358216 Inverse Möbius transform of A327936, where A327936 is multiplicative with a(p^e) = p if e >= p, otherwise 1.

1, 2, 2, 4, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 8, 2, 6, 2, 8, 4, 4, 2, 12, 3, 4, 6, 8, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 12, 2, 8, 2, 8, 6, 4, 2, 16, 3, 6, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 6, 12, 4, 8, 2, 8, 4, 8, 2, 18, 2, 4, 6, 8, 4, 8, 2, 16, 9, 4, 2, 16, 4, 4, 4, 12, 2, 12, 4, 8, 4, 4, 4, 20, 2, 6, 6, 12

Offset: 1

Author

Antti Karttunen, Nov 30 2022

Keywords

• nonn
• mult

Comments

Multiplicative because A327936 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22680
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A000005, A276086, A324655, A327936.

Programs

Mathematica

f[p_, e_] := If[e < p, e + 1, p*(e - p + 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

PARI

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
A358216(n) = sumdiv(n,d,A327936(d));

Formula

a(n) = Sum_{d|n} A327936(d).

For all n >= 0, a(A276086(n)) = A324655(n).

For all n >= 1, a(n) >= A000005(n).

Multiplicative with a(p^e) = e + 1 if e < p, and p*(e - p + 2) otherwise. - Amiram Eldar, Nov 30 2022