A293451 Number of proper divisors of n of the form 4k+1.
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, 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, 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
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
Programs
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Mathematica
a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)
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PARI
A293451(n) = sumdiv(n,d,(d
Formula
a(n) = Sum_{d|n, d
G.f.: Sum_{k>=1} x^(8*k-6) / (1 - x^(4*k-3)). - Ilya Gutkovskiy, Apr 14 2021
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