A358223 Inverse Möbius transform of A181819, prime shadow function.

1, 3, 3, 6, 3, 9, 3, 11, 6, 9, 3, 18, 3, 9, 9, 18, 3, 18, 3, 18, 9, 9, 3, 33, 6, 9, 11, 18, 3, 27, 3, 29, 9, 9, 9, 36, 3, 9, 9, 33, 3, 27, 3, 18, 18, 9, 3, 54, 6, 18, 9, 18, 3, 33, 9, 33, 9, 9, 3, 54, 3, 9, 18, 42, 9, 27, 3, 18, 9, 27, 3, 66, 3, 9, 18, 18, 9, 27, 3, 54, 18, 9, 3, 54, 9, 9, 9, 33, 3, 54

Offset: 1

Author

Antti Karttunen, Nov 30 2022

Keywords

• nonn
• mult

Comments

Multiplicative and dependent only on the prime signature (A046523) because also A181819 is.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A014284, A046523, A181819.

Programs

Mathematica

f[n_] := f[n] = Times @@ Prime@ FactorInteger[n][[All, -1]]; Array[DivisorSum[#, f] - 1 &, 90] (* Michael De Vlieger, Nov 30 2022 *)

PARI

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A358223(n) = sumdiv(n,d,A181819(d));

Formula

a(n) = Sum_{d|n} A181819(d).

Multiplicative with a(p^e) = 1 + Sum_{k=1..e} prime(k) = A014284(e+1). - Amiram Eldar, Oct 23 2023