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A359472
a(n) = 1 if the product of exponents in the prime factorization of n is 3, otherwise 0.
7
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 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, 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, 0, 0, 0, 0, 0, 1, 0, 0, 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, 0, 0, 1, 0, 0, 0, 0, 1
OFFSET
1
COMMENTS
a(n) = 1 if there is exactly one exponent in the prime factorization of n that is larger than 1, and that exponent is 3, otherwise 0.
a(n) = 1 if the number of unitary divisors of n (A034444) is equal to the number of non-unitary divisors of n (A048105), otherwise 0.
FORMULA
a(n) = [A005361(n) == 3], where [ ] is the Iverson bracket.
a(n) = [A000688(n) == 3].
a(n) = [A048106(n) == 0].
a(n) <= A359466(n) and a(n) <= A359473(n) <= A295316(n).
Sum_{k=1..n} a(k) ~ c * n, where c = (1/zeta(2)) * Sum_{k>=3} (-1)^(k+1) * P(k) = 0.0741777413..., where P is the prime zeta function. - Amiram Eldar, Jan 05 2023
MATHEMATICA
a[n_] := If[Times @@ FactorInteger[n][[;; , 2]] == 3, 1, 0]; Array[a, 100] (* Amiram Eldar, Jan 05 2023 *)
PROG
(PARI) A359472(n) = (3==factorback(factor(n)[, 2]));
(PARI) A359472(n) = (sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d))); \\ From the "is"-function in A048109 given by Michel Marcus
CROSSREFS
Characteristic function of A048109.
Differs from A295883 for the first time at n=72, where a(72) = 0, while A295883(72) = 1.
Sequence in context: A277153 A323162 A185017 * A295883 A366124 A295662
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 04 2023
STATUS
approved