A050449 a(n) = Sum_{d|n, d == 1 (mod 4)} d.

1, 1, 1, 1, 6, 1, 1, 1, 10, 6, 1, 1, 14, 1, 6, 1, 18, 10, 1, 6, 22, 1, 1, 1, 31, 14, 10, 1, 30, 6, 1, 1, 34, 18, 6, 10, 38, 1, 14, 6, 42, 22, 1, 1, 60, 1, 1, 1, 50, 31, 18, 14, 54, 10, 6, 1, 58, 30, 1, 6, 62, 1, 31, 1, 84, 34, 1, 18, 70, 6, 1, 10, 74, 38, 31, 1

Offset: 1

Author

N. J. A. Sloane, Dec 23 1999

Keywords

• nonn
• easy

Comments

Not multiplicative: a(3)*a(7) != a(21), for example. - R. J. Mathar, Dec 20 2011

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Mariusz Skałba, A Note on Sums of Two Squares and Sum-of-divisors Functions, INTEGERS 20A (2020) A92.

Crossrefs

Cf. A000593, A050452, A050460, A001826, A035451, A245058.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), this sequence (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

Programs

Maple

A050449 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if d mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050449(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

Mathematica

a[n_] := DivisorSum[n, Boole[Mod[#, 4] == 1]*#&]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2018 *)

PARI

a(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ Michel Marcus, Jan 30 2018

Formula

G.f.: Sum_{n>=0} (4*n+1)*x^(4*n+1)/(1-x^(4*n+1)). - Vladeta Jovovic, Nov 14 2002

a(n) = A000593(n) - A050452(n). - Reinhard Zumkeller, Apr 18 2006

G.f.: Sum_{n >= 1} x^n*(1 + 3*x^(4*n))/(1 - x^(4*n))^2. - Peter Bala, Dec 19 2021

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 = 0.205616... (A245058). - Amiram Eldar, Nov 26 2023

Extensions

More terms from Vladeta Jovovic, Nov 14 2002

More terms from Reinhard Zumkeller, Apr 18 2006