A359211 a(n) = tau(3*n-1)/2, where tau(n) = number of divisors of n, cf. A000005.

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 2, 3, 1, 2, 2, 5, 1, 2, 1, 3, 2, 3, 1, 4, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 1, 6, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 5, 1, 4, 2, 3, 1, 2, 1, 6, 2, 2, 2, 3, 2, 2, 2, 6, 1, 4, 1, 3, 1, 3, 3, 4, 1, 2, 1, 6, 1, 4, 1

Offset: 1

Author

Seiichi Manyama, Dec 21 2022

Keywords

• nonn
• easy

Comments

Also number of divisors of 3*n-1 of form 3*k+1 (or 3*k+2).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A078703, A099774.

Cf. A000005, A001620, A001817, A001822, A359212.

Programs

Mathematica

a[n_] := DivisorSigma[0, 3*n-1]/2; Array[a, 100] (* Amiram Eldar, Dec 21 2022 *)

PARI

a(n) = numdiv(3*n-1)/2;

PARI

a(n) = sumdiv(3*n-1, d, d%3==1);

PARI

a(n) = sumdiv(3*n-1, d, d%3==2);

PARI

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(3*k-1))))

PARI

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(3*k-2))))

Formula

G.f.: Sum_{k>0} x^k/(1 - x^(3*k-1)).

G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(3*k-2)).

Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 2*log(3))*n/3 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 26 2022