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A363805 Number of divisors of n of the form 7*k + 3.

%I A363805 #23 Nov 25 2023 04:28:26
%S A363805 0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,2,0,0,1,0,0,2,1,0,1,1,
%T A363805 0,1,0,1,1,1,0,1,0,0,2,0,0,2,0,1,2,1,0,1,0,0,1,0,1,2,0,1,1,0,0,2,0,1,
%U A363805 1,1,0,2,1,0,1,1,0,1,0,2,1,0,0,1,1,0,2,0,0,3,0,0,2,1,0,2,0,0,1
%N A363805 Number of divisors of n of the form 7*k + 3.
%H A363805 Amiram Eldar, <a href="/A363805/b363805.txt">Table of n, a(n) for n = 1..10000</a>
%H A363805 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 A363805 G.f.: Sum_{k>0} x^(3*k)/(1 - x^(7*k)).
%F A363805 G.f.: Sum_{k>0} x^(7*k-4)/(1 - x^(7*k-4)).
%F A363805 Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,7) - (1 - gamma)/7 = -0.0004108181..., gamma(3,7) = -(psi(3/7) + 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 A363805 a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 3 &]; Array[a, 100] (* _Amiram Eldar_, Jun 23 2023 *)
%o A363805 (PARI) a(n) = sumdiv(n, d, d%7==3);
%Y A363805 Cf. A279061, A363795, A363806, A363807, A363808.
%Y A363805 Cf. A001842, A001878.
%Y A363805 Cf. A284444.
%Y A363805 Cf. A001620, A016630, A354629 (psi(3/7)).
%K A363805 nonn,easy
%O A363805 1,24
%A A363805 _Seiichi Manyama_, Jun 23 2023