This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188171 #23 Nov 25 2023 04:37:19
%S A188171 0,0,0,0,1,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0,0,0,
%T A188171 1,0,1,0,1,1,0,1,0,0,2,0,0,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,2,0,0,0,
%U A188171 1,1,0,0,0,1,1,0,1,1,0,1,0,0,0,1,2,0,1,0,0,2
%N A188171 The number of divisors d of n of the form d == 5 (mod 8).
%C A188171 a(5n) >= 1 as d=5 contributes to the count.
%H A188171 Antti Karttunen, <a href="/A188171/b188171.txt">Table of n, a(n) for n = 1..10000</a>
%H A188171 Michael D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H A188171 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 A188171 A188169(n)+a(n) = A001826(n).
%F A188171 A188169(n)+A188170(n)-a(n)-A188172(n) = A002325(n).
%F A188171 G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(8*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A188171 Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,8) - (1 - gamma)/8 = -0.131189..., gamma(5,8) = -(psi(5/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%e A188171 a(13) = 1 because the divisor d=13 is 8+5 == 5 (mod 8).
%p A188171 sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
%p A188171 A188171 := proc(n) sigmamr(n,8,5) ; end proc:
%t A188171 a[n_] := DivisorSum[n, 1 &, Mod[#, 8] == 5 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A188171 (PARI) A188171(n) = sumdiv(n, d, (5==(d%8))); \\ _Antti Karttunen_, Jul 09 2017
%Y A188171 Cf. A001826, A002325, A188169, A188170, A188172.
%Y A188171 Cf. A001620, A016631, A354634 (psi(5/8)).
%K A188171 nonn,easy
%O A188171 1,45
%A A188171 _R. J. Mathar_, Mar 23 2011