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

%I #20 Nov 25 2023 08:03:13

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

%H Antti Karttunen, <a href="/A293513/b293513.txt">Table of n, a(n) for n = 1..16384</a>

%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 a(n) = Sum_{d|n, d<n} [3 == d mod 4].

%F a(n) = A091954(n) - A293451(n).

%F a(n) = A001842(n) - A121262(n+1).

%F G.f.: Sum_{k>=1} x^(8*k-2) / (1 - x^(4*k-1)). - _Ilya Gutkovskiy_, Apr 14 2021

%F 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 a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 3 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)

%o (PARI) A293513(n) = sumdiv(n,d,(d<n)*(3==(d%4)));

%Y Cf. A001842, A091954, A121262, A293451, A293903.

%K nonn,easy

%O 1,21

%A _Antti Karttunen_, Oct 19 2017