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A363807 Number of divisors of n of the form 7*k + 5.

%I A363807 #21 Nov 25 2023 04:29:11
%S A363807 0,0,0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,1,0,0,1,0,
%T A363807 1,1,0,1,0,2,0,0,0,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,2,1,0,0,0,1,1,0,1,
%U A363807 0,1,0,1,0,0,2,1,0,1,0,2,0,1,0,1,1,0,0,0,1,1,0,0,0,1,2,2,0,0,1,1
%N A363807 Number of divisors of n of the form 7*k + 5.
%H A363807 Amiram Eldar, <a href="/A363807/b363807.txt">Table of n, a(n) for n = 1..10000</a>
%H A363807 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 A363807 G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k)).
%F A363807 G.f.: Sum_{k>0} x^(7*k-2)/(1 - x^(7*k-2)).
%F A363807 Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,7) - (1 - gamma)/7 = -0.169787..., gamma(5,7) = -(psi(5/7) + 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 A363807 a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* _Amiram Eldar_, Jun 23 2023 *)
%o A363807 (PARI) a(n) = sumdiv(n, d, d%7==5);
%Y A363807 Cf. A279061, A363795, A363805, A363806, A363808.
%Y A363807 Cf. A284446, A319995.
%Y A363807 Cf. A001620, A016630, A354631 (psi(5/7)).
%K A363807 nonn,easy
%O A363807 1,40
%A A363807 _Seiichi Manyama_, Jun 23 2023