|
%I #22 Nov 25 2023 04:40:54
%S 0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,1,0,0,0,
%T 1,0,0,0,1,0,0,1,0,0,1,1,1,0,1,0,0,0,0,0,1,1,0,0,0,1,0,1,2,0,0,0,0,0,
%U 1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,1,0,0,1,0,0,1
%N Number of divisors d of n of the form d == 7 (mod 8).
%H Reinhard Zumkeller, <a href="/A188172/b188172.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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(n)+a(n) = A001842(n).
%F A188169(n)+A188170(n)-A188171(n)-a(n) = A002325(n).
%F a(A188226(n))=n and a(m)<>n for m<A188226(n), n>=0; a(A141164(n))=1. - _Reinhard Zumkeller_, Mar 26 2011
%F G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(8*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,8) - (1 - gamma)/8 = -0.212276..., gamma(7,8) = -(psi(7/8) + 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 a(A007522(i)) = 1, any i.
%p 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 A188172 := proc(n) sigmamr(n,8,7) ; end proc:
%t Table[Count[Divisors[n],_?(Mod[#,8]==7&)],{n,90}] (* _Harvey P. Dale_, Mar 08 2014 *)
%o (Haskell)
%o a188172 n = length $ filter ((== 0) . mod n) [7,15..n]
%o -- _Reinhard Zumkeller_, Mar 26 2011
%o (PARI) a(n) = sumdiv(n, d, (d % 8) == 7); \\ _Amiram Eldar_, Nov 25 2023
%Y Cf. A001842, A002325, A004771, A141164, A188169, A188170, A188171, A188226.
%Y Cf. A001620, A016631, A354635 (psi(7/8)).
%K nonn,easy
%O 1,63
%A _R. J. Mathar_, Mar 23 2011
|
