A369070 - OEIS

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A369070

a(n) = 1 if there is at least one prime power p^e in the prime factorization of n such that p|e, otherwise 0.

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

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OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

For n >= 1, a(n) <= A342023(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Product_{p prime} (1 - (p - 1)/(p*(p^p - 1))) = 0.18824296270011399086... . - Amiram Eldar, Jan 15 2024

MAPLE

a:= n-> `if`(ormap(i-> irem(i[2], i[1])=0, ifactors(n)[2]), 1, 0):

seq(a(n), n=1..124); # Alois P. Heinz, Jan 15 2024

MATHEMATICA

a[n_] := If[AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &], 1, 0]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 15 2024 *)

PROG

(PARI) A369070(n) = { my(f=factor(n)); for(i=1, #f~, if(!(f[i, 2]%f[i, 1]), return(1))); (0); };

(SageMath)

def isA369070(n): return any(f[1] % f[0] == 0 for f in factor(n))

print([int(isA369070(n)) for n in range(1, 101)]) # Peter Luschny, Jan 16 2024

CROSSREFS

Characteristic function of A342090.