A351114 Characteristic function of balanced numbers: a(n) = 1 if phi(n) divides sigma(n), otherwise 0.

1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1

Comments

A balanced number k is a number such that phi(k) | sigma(k).

If a(x) = 1, a(y) = 1, and gcd(x,y) = 1, then a(x*y) = 1 also. - Antti Karttunen, Jan 01 2023, based on Enrique Pérez Herrero's Sep 05 2010 comment in A020492.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100130

Index entries for characteristic functions

Formula

a(n) = c(sigma(n)/phi(n)), where c(n) = 1 - ceiling(n) + floor(n).

a(n) = [A063514(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 01 2023

Mathematica

a[n_] := Boole[Divisible[DivisorSigma[1, n], EulerPhi[n]]]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)

Prog

(Python)
from math import prod
from sympy import factorint
def A351114(n):
    f = factorint(n)
    return int(not prod(p*(p**(e+1)-1) for p, e in f.items()) % (n*prod((p-1)**2 for p in f))) # Chai Wah Wu, Feb 01 2022
(PARI) A351114(n) = !(sigma(n)%eulerphi(n)); \\ Antti Karttunen, Jan 01 2023

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A063514.

Sequence in context: A204443 A105349 A096606 * A128190 A128189 A303340

Adjacent sequences: A351111 A351112 A351113 * A351115 A351116 A351117

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 31 2022

Extensions

Data section extended up to a(105) and the name amended with a formula by Antti Karttunen, Jan 01 2023

Status

approved