A001842 Expansion of Sum_{n>=0} x^(4*n+3)/(1 - x^(4*n+3)).

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 1, 2, 0, 0, 2, 1, 1, 2, 1, 1, 1, 1, 0, 2, 0, 0, 2, 2, 1, 2, 0, 1, 2, 0, 1, 3, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 0, 3, 1, 2, 2, 1, 0, 2, 0, 1, 2, 0, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of divisors of n of the form 4*k+3. - Reinhard Zumkeller, Apr 18 2006

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 244.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Michael Gilleland, Some Self-Similar Integer Sequences.
• 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.

Crossrefs

Cf. A001227, A001620, A001826, A072437, A256846.

Programs

Maple

with(numtheory): seq(add(binomial(d,3) mod 2, d in divisors(n)), n=0..100); # Ridouane Oudra, Nov 19 2019

Mathematica

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 4] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 4] == 3 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

PARI

a(n) = if(n<1, 0, sumdiv(n, d, d%4 == 3)); \\ Amiram Eldar, Nov 25 2023

Formula

a(A072437(n)) = 0. - Benoit Cloitre, Apr 24 2003

a(n) = A001227(n) - A001826(n). - Reinhard Zumkeller, Apr 18 2006

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 11 2019

a(n) = Sum_{d|n} (binomial(d,3) mod 2). - Ridouane Oudra, Nov 19 2019

Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (1 - gamma)/4 = A256846 - (1 - A001620)/4 = -0.180804... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023