A363805 Number of divisors of n of the form 7*k + 3.
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, 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, 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
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
Crossrefs
Programs
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Mathematica
a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)
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PARI
a(n) = sumdiv(n, d, d%7==3);
Formula
G.f.: Sum_{k>0} x^(3*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-4)/(1 - x^(7*k-4)).
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