cp's OEIS Frontend

A293513 Number of proper divisors of n of the form 4k+3.

%I A293513 #20 Nov 25 2023 08:03:13
%S A293513 0,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,2,1,0,1,0,0,1,1,0,2,0,0,2,0,
%T A293513 1,1,0,1,1,0,0,2,0,1,2,1,0,1,1,0,1,0,0,2,1,1,2,0,0,2,0,1,2,0,0,2,0,0,
%U A293513 2,2,0,1,0,0,2,1,2,2,0,0,2,0,0,2,0,1,1,1,0,2,1,1,2,1,1,1,0,1,2,0,0,2,0,0,4
%N A293513 Number of proper divisors of n of the form 4k+3.
%H A293513 Antti Karttunen, <a href="/A293513/b293513.txt">Table of n, a(n) for n = 1..16384</a>
%H A293513 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 A293513 a(n) = Sum_{d|n, d<n} [3 == d mod 4].
%F A293513 a(n) = A091954(n) - A293451(n).
%F A293513 a(n) = A001842(n) - A121262(n+1).
%F A293513 G.f.: Sum_{k>=1} x^(8*k-2) / (1 - x^(4*k-1)). - _Ilya Gutkovskiy_, Apr 14 2021
%F A293513 Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (2 - gamma)/4 = A256846 - (2 - A001620)/4 = -0.430804... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%t A293513 a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 3 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A293513 (PARI) A293513(n) = sumdiv(n,d,(d<n)*(3==(d%4)));
%Y A293513 Cf. A001842, A091954, A121262, A293451, A293903.
%K A293513 nonn,easy
%O A293513 1,21
%A A293513 _Antti Karttunen_, Oct 19 2017