A001822 Expansion of Sum_{n>=0} x^(3n+2)/(1-x^(3n+2)).

0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 0, 3, 1, 2, 2, 1, 0, 2, 0, 4, 1, 2, 0, 3, 1, 2, 1, 2, 0, 3, 1, 2, 1, 1, 2, 4, 0, 2, 1, 3, 0, 2, 0, 3, 2, 2, 0, 3, 1, 4, 1, 2, 0, 2, 1, 2, 2, 2, 0, 5, 0, 2, 1, 2, 2, 2, 1, 4, 1, 2, 0, 3, 0, 2, 2, 3, 0, 3, 1, 4, 1, 2, 0, 4, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of positive divisors of n of the form 3k+2. If r(n) denotes the number of representations of n by the quadratic form j^2+i*j+i^2, then r(n)= 6 *(A001817(n)-a(n)). - Benoit Cloitre, Jun 24 2002

References

• Bruce C. Berndt,"On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992),33-43.

Links

• Nick Hobson, Table of n, a(n) for n = 1..10000
• P. G. Dirichlet, Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 21 (1840), 1-12.
• 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. A000005, A001620, A001817, A002324, A035191, A256843.

Programs

Haskell

a001822 n = length [d | d <- [2,5..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 26 2011

Maple

A001822 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) = 2 then
            a := a+1 ;
        end if ;
    end do:
    a ;
end proc:
seq(A001822(n),n=1..100) ; # R. J. Mathar, Sep 25 2017

Mathematica

a[n_] := DivisorSum[n, Boole[Mod[#, 3] == 2]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

PARI

a(n)=if(n<1, 0, sumdiv(n,d, d%3==2))

Formula

Moebius transform is period 3 sequence [0, 1, 0, ...]. - Michael Somos, Sep 20 2005

G.f.: Sum_{k>0} x^(3k-1)/(1-x^(3k-1)) = Sum_{k>0} x^(2k)/(1-x^(3k)). - Michael Somos, Sep 20 2005

a(n) = (A035191(n) - A002324(n)) / 2. - Reinhard Zumkeller, Nov 26 2011

a(n) + A001817(n) + A000005(n/3) = A000005(n), where A000005(.)=0 if the argument is not an integer. - R. J. Mathar, Sep 25 2017

Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (1 - gamma)/3 = A256843 - (1 - A001620)/3 = -0.0677207... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023