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A188170 The number of divisors d of n of the form d == 3 (mod 8).

%I A188170 #26 Nov 25 2023 04:36:59
%S A188170 0,0,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,2,0,0,1,0,0,2,0,
%T A188170 1,1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,2,0,0,2,1,0,2,0,1,1,0,0,1,0,0,2,1,0,
%U A188170 1,1,0,1,0,0,2,1,1,1,0,0,2,0,1,1,0,1,1,1,0,1
%N A188170 The number of divisors d of n of the form d == 3 (mod 8).
%C A188170 a(3n) >= 1 as the divisor d=3 contributes to the count then.
%H A188170 Michael De Vlieger, <a href="/A188170/b188170.txt">Table of n, a(n) for n = 1..10000</a>
%H A188170 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 A188170 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 A188170 a(n) + A188172(n) = A001842(n).
%F A188170 A188169(n) + a(n) - A188171(n) - A188172(n) = A002325(n).
%F A188170 G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(8*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A188170 Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,8) - (1 - gamma)/8 = A256782 - (1 - A001620)/8 = 0.0314716... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%p A188170 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 A188170 A188170 := proc(n) sigmamr(n,8,3) ; end proc:
%t A188170 Table[Count[Divisors[n],_?(Mod[#,8]==3&)],{n,100}] (* _Harvey P. Dale_, Jul 08 2013 *)
%o A188170 (PARI) a(n) = sumdiv(n, d, (d%8) == 3); \\ _Michel Marcus_, Nov 05 2018
%Y A188170 Cf. A001842, A002325, A188169, A188171, A188172.
%Y A188170 Cf. A001620, A256782.
%K A188170 nonn,easy
%O A188170 1,27
%A A188170 _R. J. Mathar_, Mar 23 2011