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

1,10

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.

Formula

a(n) = Sum_{d|n, d<n} [1 == d mod 4].

a(n) = A091954(n) - A293513(n).

a(n) = A001826(n) - A121262(n-1).

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

Mathematica

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

Prog

(PARI) A293451(n) = sumdiv(n, d, (d<n)*(1==(d%4)));

Crossrefs

Cf. A001620, A001826, A091954, A121262, A256778, A293513, A293901.

Sequence in context: A157196 A300410 A362845 * A063014 A286361 A357849

Adjacent sequences: A293448 A293449 A293450 * A293452 A293453 A293454

Keyword

nonn,easy

Author

Antti Karttunen, Oct 19 2017

Status

approved