A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

1, 1, 1, 7, 1, 1, 1, 15, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 15, 31, 1, 40, 7, 1, 1, 1, 63, 1, 1, 1, 91, 1, 1, 1, 15, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 40, 1, 15, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 195, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 15, 1, 13, 1, 7, 1, 1, 1, 63

Offset: 1

Author

Antti Karttunen, Nov 25 2017

Keywords

• nonn
• easy
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000203, A001694, A003557, A005117, A057521, A064549, A092261, A295295.

Programs

Mathematica

Array[DivisorSigma[1, #/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] ] &, 96] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 08 2022 *)

PARI

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ Amiram Eldar, Oct 08 2022

Python

from math import prod
from sympy import factorint
def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # Chai Wah Wu, Nov 14 2022

Scheme

(define (A295294 n) (A000203 (A057521 n)))
;; With memoization-macro definec:
(definec (A295294 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= e 1) 1 (/ (- (expt p (+ 1 e)) 1) (- p 1))) (A295294 (A028234 n))))))

Formula

Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.

a(n) = A000203(A057521(n)) = A000203(A064549(A003557(n))).

a(n) = A000203(n) / A092261(n).

From Amiram Eldar, Oct 08 2022: (Start)

a(n) = 1 iff n is squarefree (A005117).

a(n) = A000203(n) iff n is powerful (A001694). (End)

Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 09 2023