A351568 Sum of the divisors of the largest unitary divisor of n that is a square.

1, 1, 1, 7, 1, 1, 1, 1, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 1, 31, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 91, 1, 1, 1, 1, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 1, 1, 13, 1, 7, 1, 1, 1, 1, 1, 57, 13

Offset: 1

Author

Antti Karttunen, Feb 23 2022

Keywords

• nonn
• mult

Comments

Obviously, all terms are odd.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A002117, A350388, A351569, A351570, A351575 (positions of primes).

Programs

Mathematica

f[p_, e_] := If[EvenQ[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

A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));

Python

from math import prod
from sympy import factorint
def A351568(n): return prod(1 if e % 2 else (p**(e+1)-1)//(p-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 even and 1 otherwise.

a(n) = A000203(A350388(n)).

a(n) = A000203(n) / A351569(n).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) + 1/p^2 - 1/p^(5/2)) = 1.008259499413... . - Amiram Eldar, Nov 20 2022

Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 03 2023