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A363806 Number of divisors of n of the form 7*k + 4.

%I A363806 #20 Nov 25 2023 04:28:48
%S A363806 0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0,0,2,1,0,
%T A363806 0,2,0,0,1,1,0,0,0,2,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,2,0,0,0,2,0,1,1,1,
%U A363806 0,0,0,2,0,1,1,1,1,1,0,1,1,0,0,1,0,0,0,3,0,1,0,2,0,0,1,2,0,0,1,2,0
%N A363806 Number of divisors of n of the form 7*k + 4.
%H A363806 Amiram Eldar, <a href="/A363806/b363806.txt">Table of n, a(n) for n = 1..10000</a>
%H A363806 R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
%F A363806 G.f.: Sum_{k>0} x^(4*k)/(1 - x^(7*k)).
%F A363806 G.f.: Sum_{k>0} x^(7*k-3)/(1 - x^(7*k-3)).
%F A363806 Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,7) - (1 - gamma)/7 = -0.102846..., gamma(4,7) = -(psi(4/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%t A363806 a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 4 &]; Array[a, 100] (* _Amiram Eldar_, Jun 23 2023 *)
%o A363806 (PARI) a(n) = sumdiv(n, d, d%7==4);
%Y A363806 Cf. A279061, A363795, A363805, A363807, A363808.
%Y A363806 Cf. A284445.
%Y A363806 Cf. A001620, A016630, A354630 (psi(4/7)).
%K A363806 nonn,easy
%O A363806 1,32
%A A363806 _Seiichi Manyama_, Jun 23 2023