A359305 Number of divisors of 6*n-1 of form 6*k+1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 3, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 3

Offset: 1

Author

Seiichi Manyama, Dec 25 2022

Keywords

• nonn
• easy

Comments

Also number of divisors of 6*n-1 of form 6*k+5.

Links

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

Crossrefs

Cf. A279060, A319995.

Cf. A000005, A001620, A016969, A078703, A359211, A359233.

Cf. A359306, A359307, A359308, A359309.

Cf. A359324, A359325, A359326, A359327.

Programs

Mathematica

a[n_] := DivisorSigma[0, 6*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

PARI

a(n) = sumdiv(6*n-1, d, d%6==1);

PARI

a(n) = sumdiv(6*n-1, d, d%6==5);

PARI

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-1))))

PARI

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(5*k-4)/(1-x^(6*k-5))))

Formula

a(n) = A279060(6*n-1) = A319995(6*n-1).

G.f.: Sum_{k>0} x^k/(1 - x^(6*k-1)).

G.f.: Sum_{k>0} x^(5*k-4)/(1 - x^(6*k-5)).

From Amiram Eldar, Dec 26 2022: (Start)

a(n) = A000005(A016969(n-1))/2.

Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 3*log(2) + 2*log(3))*n/6 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)