A371413 Dedekind psi function applied to the cubefull numbers (A036966).

1, 12, 24, 36, 48, 96, 108, 150, 192, 432, 324, 384, 392, 864, 768, 750, 1296, 972, 1728, 1800, 1536, 2592, 1452, 3456, 3888, 3600, 3072, 2916, 2366, 2744, 5184, 4704, 3750, 5400, 6912, 7776, 7200, 6144, 5202, 9000, 10368, 9408, 11664, 8748, 7220, 13824, 15552

Offset: 1

Author

Amiram Eldar, Mar 22 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001615, A036966, A065464.

Similar sequences: A323332, A371412, A371415.

Programs

Mathematica

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p+1) * p^(e-1)), Nothing]]]; Array[f, 20000]

PARI

dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(dedpsi(f), ", "))); }

Formula

a(n) = A001615(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p^2-1)*p)) = 1.231291... (A065487).