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Number of divisors of 6*n-1 of form 6*k+1.
9

%I #21 Dec 27 2022 07:08:28

%S 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,

%T 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,

%U 2,1,3,1,2,1,1,4,1,1,2,1,2,1,2,1,1,2,1,2,2,3

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

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

%H Seiichi Manyama, <a href="/A359305/b359305.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F From _Amiram Eldar_, Dec 26 2022: (Start)

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

%F 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)

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

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

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

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

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

%Y Cf. A279060, A319995.

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

%Y Cf. A359306, A359307, A359308, A359309.

%Y Cf. A359324, A359325, A359326, A359327.

%K nonn,easy

%O 1,6

%A _Seiichi Manyama_, Dec 25 2022