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A355825
a(n) = 1 if all exponents in prime factorization of n have an odd binary weight, otherwise 0.
5
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
OFFSET
1
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
KEYWORD
nonn,easy,mult
AUTHOR
Antti Karttunen, Jul 19 2022
STATUS
approved