A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

1, 3, 4, 1, 6, 12, 8, 15, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 60, 1, 42, 40, 8, 30, 72, 32, 63, 48, 54, 48, 1, 38, 60, 56, 90, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 120, 72, 120, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84

Offset: 1

Author

Antti Karttunen, Feb 23 2022

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A013662, A028982 (positions of odd terms), A268335 (exponentially odd numbers), A350389, A351568, A351571.

Coincides with A001615 on squarefree numbers, A005117.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

PARI

A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351569(n) = sigma(A350389(n));

Python

from math import prod
from sympy import factorint
def A351569(n): return prod((p**(e+1)-1)//(p-1) if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Formula

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and 1 otherwise.

a(n) = A000203(A350389(n)).

a(n) = A000203(n) / A351568(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(4)/2 = Pi^4/180 = 0.541161... . - Amiram Eldar, Nov 20 2022

Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s - 1/p^(2*s-2)). - Amiram Eldar, Sep 03 2023