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A293451 Number of proper divisors of n of the form 4k+1.

%I A293451 #19 Nov 25 2023 08:02:16
%S A293451 0,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,2,1,2,1,1,1,1,2,2,2,1,1,2,1,1,1,2,
%T A293451 2,2,1,1,2,2,1,2,1,1,3,1,1,1,1,3,2,2,1,2,2,1,1,2,1,2,1,1,3,1,3,2,1,2,
%U A293451 1,2,1,2,1,2,3,1,1,2,1,2,2,2,1,2,3,1,2,1,1,4,2,1,1,1,2,1,1,2,3,3,1,2,1,2,3
%N A293451 Number of proper divisors of n of the form 4k+1.
%H A293451 Antti Karttunen, <a href="/A293451/b293451.txt">Table of n, a(n) for n = 1..16384</a>
%H A293451 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 A293451 a(n) = Sum_{d|n, d<n} [1 == d mod 4].
%F A293451 a(n) = A091954(n) - A293513(n).
%F A293451 a(n) = A001826(n) - A121262(n-1).
%F A293451 G.f.: Sum_{k>=1} x^(8*k-6) / (1 - x^(4*k-3)). - _Ilya Gutkovskiy_, Apr 14 2021
%F A293451 Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,4) - (2 - gamma)/4 = A256778 - (2 - A001620)/4 = 0.354593... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%t A293451 a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A293451 (PARI) A293451(n) = sumdiv(n,d,(d<n)*(1==(d%4)));
%Y A293451 Cf. A001620, A001826, A091954, A121262, A256778, A293513, A293901.
%K A293451 nonn,easy
%O A293451 1,10
%A A293451 _Antti Karttunen_, Oct 19 2017