A359472 - OEIS

%I #26 Jan 05 2023 03:20:14

%S 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,

%T 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,

%U 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

%N a(n) = 1 if the product of exponents in the prime factorization of n is 3, otherwise 0.

%C 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.

%C 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.

%H Antti Karttunen, <a href="/A359472/b359472.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = [A005361(n) == 3], where [ ] is the Iverson bracket.

%F a(n) = [A000688(n) == 3].

%F a(n) = [A048106(n) == 0].

%F a(n) <= A359466(n) and a(n) <= A359473(n) <= A295316(n).

%F 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

%t a[n_] := If[Times @@ FactorInteger[n][[;; , 2]] == 3, 1, 0]; Array[a, 100] (* _Amiram Eldar_, Jan 05 2023 *)

%o (PARI) A359472(n) = (3==factorback(factor(n)[, 2]));

%o (PARI) A359472(n) = (sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d))); \\ From the "is"-function in A048109 given by _Michel Marcus_

%Y Characteristic function of A048109.

%Y Cf. A000688, A005361, A034444, A048105, A048106, A295316, A359466, A359471, A359473, A359474.

%Y Differs from A295883 for the first time at n=72, where a(72) = 0, while A295883(72) = 1.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 04 2023