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

1,2

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

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

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 *)

Prog

(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));

Crossrefs

Cf. A000005, A276086, A324655, A327936.

Sequence in context: A147848 A306652 A239614 * A193432 A129089 A255201

Adjacent sequences: A358213 A358214 A358215 * A358217 A358218 A358219

Keyword

nonn,mult

Author

Antti Karttunen, Nov 30 2022

Status

approved