A380326 a(n) is the sum of squarefree divisors of the n-th exponentially odd number.

1, 3, 4, 6, 12, 8, 3, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 12, 42, 4, 30, 72, 32, 3, 48, 54, 48, 38, 60, 56, 18, 42, 96, 44, 72, 48, 72, 54, 12, 72, 24, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 36, 90, 112

Offset: 1

Author

Amiram Eldar, Jan 20 2025

Keywords

• nonn
• easy

Comments

The number of squarefree divisors of the n-th exponentially odd number is A366534(n).

Links

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

Crossrefs

Cf. A005117, A048250, A065463, A268335, A366439, A366534, A366535.

Programs

Mathematica

f[p_, e_] := p + 1; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

PARI

A048250(f) = vecprod(apply(x -> x+1, f[, 1]));
list(lim) = for(k = 1, lim, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(A048250(f), ", ")));

Formula

a(n) = A048250(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} (1 - 1/(p*(p^2+1))) = 1.73161118511498727822..., and d = A065463 is the asymptotic density of the exponentially odd numbers.