cp's OEIS Frontend

A115069 a(n) = 3^b(n), where b(n) is #{primes p=1 mod 3 dividing n}.

%I A115069 #22 Nov 30 2024 06:27:02
%S A115069 1,1,1,1,1,1,3,1,1,1,1,1,3,3,1,1,1,1,3,1,3,1,1,1,1,3,1,3,1,1,3,1,1,1,
%T A115069 3,1,3,3,3,1,1,3,3,1,1,1,1,1,3,1,1,3,1,1,1,3,3,1,1,1,3,3,3,1,3,1,3,1,
%U A115069 1,3,1,1,3,3,1,3,3,3,3,1,1,1,1,3,1,3,1,1,1,1,9,1,3,1,3,1,3,3,1
%N A115069 a(n) = 3^b(n), where b(n) is #{primes p=1 mod 3 dividing n}.
%H A115069 Amiram Eldar, <a href="/A115069/b115069.txt">Table of n, a(n) for n = 1..10000</a>
%H A115069 Steven Finch and Pascal Sebah, <a href="https://arxiv.org/abs/math/0604465">Squares and Cubes Modulo n</a>, arXiv:math/0604465 [math.NT], 2006-2016.
%F A115069 a(n) = 3^A005088(n). - _R. J. Mathar_, May 19 2020
%F A115069 From _Amiram Eldar_, Nov 30 2024: (Start)
%F A115069 Multiplicative with a(p^e) = 3 if p == 1 (mod 3), and 1 otherwise.
%F A115069 Sum_{k=1..n} a(k) ~ (sqrt(3)/(2*Pi)) * c * n * log(n), where c = Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))) = 0.9410349413195354517900322... (Finch and Sebah, 2006). (End)
%p A115069 a:= n-> 3^add(`if`(irem(i[1], 3)=1, 1, 0), i=ifactors(n)[2](n)):
%p A115069 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 17 2019
%t A115069 b[n_] := Count[FactorInteger[n][[All, 1]], p_ /; Mod[p, 3] == 1];
%t A115069 a[1] = 1; a[n_] := 3^b[n];
%t A115069 Table[a[n], {n, 1, 99}] (* _Jean-François Alcover_, Feb 17 2019 *)
%o A115069 (PARI) a(n) = 3^#select(x -> x%3 == 1, factor(n)[,1]); \\ _Amiram Eldar_, Nov 30 2024
%Y A115069 Cf. A005088, A060839.
%K A115069 nonn,mult
%O A115069 1,7
%A A115069 _Steven Finch_, Mar 01 2006
%E A115069 a(1)=1 prepended by _Alois P. Heinz_, Feb 17 2019
%E A115069 Keyword mult added by _Amiram Eldar_, Nov 30 2024