A355825 - OEIS

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A355825

a(n) = 1 if all exponents in prime factorization of n have an odd binary weight, otherwise 0.

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

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OFFSET

LINKS

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

Index entries for characteristic functions.

Index entries for sequences computed from exponents in factorization of n.

FORMULA

Multiplicative with a(p^e) = A010060(e).

For all n >= 1, a(n) >= A355823(n) >= A302777(n).

a(n) = A295316(A268385(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + f(1/p)) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023

MATHEMATICA

a[n_] := If[AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)

PROG

(PARI) A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));

CROSSREFS

Characteristic function of A270428 (exponentially odious numbers).

Cf. A000069, A010060, A268385, A295316, A302777, A355826 (Dirichlet inverse).

Differs from related A355823 for the first time at n=128, where a(128) = 1, while A355823(128) = 0.

Cf. also A270419.

Sequence in context: A330548 A225817 A355823 * A332732 A248863 A328306

Adjacent sequences: A355822 A355823 A355824 * A355826 A355827 A355828

KEYWORD

nonn,easy,mult

AUTHOR

Antti Karttunen, Jul 19 2022

STATUS

approved