A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

1, 2, 3, 4, 6, 6, 7, 8, 10, 12, 11, 12, 14, 14, 18, 16, 18, 20, 19, 24, 22, 22, 23, 24, 31, 28, 30, 28, 30, 36, 31, 32, 34, 36, 42, 40, 38, 38, 42, 48, 42, 44, 43, 44, 60, 46, 47, 48, 50, 62, 54, 56, 54, 60, 66, 56, 58, 60, 59, 72, 62, 62, 73, 64, 84, 68

Offset: 1

Author

N. J. A. Sloane, Dec 23 1999

Keywords

• nonn
• easy

Comments

Not multiplicative: a(3)*a(7) <> a(21), for example.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001826, A002131, A050449, A050464, A050469, A222183.

Cf. A050461, A050462, A050463.

Programs

Maple

A050460 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050460(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

Mathematica

a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#&]; Array[a, 70] (* Jean-François Alcover, Dec 01 2015 *)

PARI

a(n)=sumdiv(n,d,if(n/d%4==1,d)) \\ Charles R Greathouse IV, Dec 04 2013

Formula

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

G.f.: Sum_{k>0} x^(4*k-3) / (1 - x^(4*k-3))^2. - Seiichi Manyama, Jun 29 2023

from Amiram Eldar, Nov 05 2023: (Start)

a(n) = A002131(n) - A050464(n).

a(n) = A050469(n) + A050464(n).

a(n) = (A002131(n) + A050469(n))/2.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A222183. (End)