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

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, 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, 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

Offset: 1

Author

Steven Finch, Mar 01 2006

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Steven Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Crossrefs

Cf. A005088, A060839.

Programs

Maple

a:= n-> 3^add(`if`(irem(i[1], 3)=1, 1, 0), i=ifactors(n)[2](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 17 2019

Mathematica

b[n_] := Count[FactorInteger[n][[All, 1]], p_ /; Mod[p, 3] == 1];
a[1] = 1; a[n_] := 3^b[n];
Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Feb 17 2019 *)

PARI

a(n) = 3^#select(x -> x%3 == 1, factor(n)[,1]); \\ Amiram Eldar, Nov 30 2024

Formula

a(n) = 3^A005088(n). - R. J. Mathar, May 19 2020

From Amiram Eldar, Nov 30 2024: (Start)

Multiplicative with a(p^e) = 3 if p == 1 (mod 3), and 1 otherwise.

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)

Extensions

a(1)=1 prepended by Alois P. Heinz, Feb 17 2019

Keyword mult added by Amiram Eldar, Nov 30 2024