A082051 -id:A082051 - cp's OEIS frontend

A082050 Sum of divisors of n that are not of the form 3k+1.

0, 2, 3, 2, 5, 11, 0, 10, 12, 7, 11, 23, 0, 16, 23, 10, 17, 38, 0, 27, 24, 13, 23, 55, 5, 28, 39, 16, 29, 61, 0, 42, 47, 19, 40, 86, 0, 40, 42, 35, 41, 88, 0, 57, 77, 25, 47, 103, 0, 57, 71, 28, 53, 119, 16, 80, 60, 31, 59, 153, 0, 64, 96, 42, 70, 121, 0, 87, 95, 56, 71, 190

Offset: 1

Ralf Stephan, Apr 02 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A004611, A027748, A046913, A078181, A078182, A082051, A086463.

sd[n_]:=Total[Select[Divisors[n],!IntegerQ[(#-1)/3]&]]; Array[sd,80] (* Harvey P. Dale, May 04 2011 *)

for(n=1,100,print1(sumdiv(n,d,if(d%3!=1,d))","))

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, (3*n-1)*x^(3*n-1)/(1-x^(3*n-1)) + (3*n)*x^(3*n)/(1-x^(3*n)) );
v = Vec(gf);  concat([0],v)
\\ Joerg Arndt, May 17 2013

a(A004611(n)) = 0.
G.f.: Sum_{k>=1} x^(2*k)*(2+3*x^k+x^(3*k))/(1-x^(3*k))^2. - Vladeta Jovovic, Apr 11 2006
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Jan 06 2024

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035360, A035386, A081362, A082051, A132463, A137569, A274719, A284312, A374060.

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 2)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082051(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035360(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035386(n-k).

A326394 Expansion of Sum_{k>=1} x^k * (1 + x^(2k)) / (1 - x^(3k)).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 2, 2, 3, 3, 1, 5, 2, 3, 4, 2, 1, 6, 2, 2, 4, 4, 1, 6, 2, 3, 3, 2, 2, 8, 2, 2, 4, 4, 1, 6, 2, 3, 5, 2, 1, 8, 3, 3, 3, 4, 1, 7, 2, 4, 4, 2, 1, 9, 2, 2, 6, 4, 2, 6, 2, 3, 3, 4, 1, 10, 2, 2, 5, 4, 2, 6, 2, 5, 5, 2, 1, 10, 2, 2, 3, 4, 1, 10, 4

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Number of divisors of n that are not of the form 3*k + 2.

Cf. A000005, A001817, A001822, A003627, A032766, A035191, A082051, A326395.

Cf. A001620, A256843.

nmax = 90; CoefficientList[Series[Sum[x^k (1 + x^(2 k))/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, 1 &, !MemberQ[{2}, Mod[#, 3]] &], {n, 1, 90}]

a(n) = {numdiv(n) - sumdiv(n, d, d%3==2)} \\ Andrew Howroyd, Sep 11 2019

a(n) = A000005(n) - A001822(n).

Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(2,3) = (5*A001620-2)/3 - A256843 = 0.222152... . - Amiram Eldar, Jan 14 2024

cp's OEIS Frontend

A082050 Sum of divisors of n that are not of the form 3k+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A326394 Expansion of Sum_{k>=1} x^k * (1 + x^(2k)) / (1 - x^(3k)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A082050 Sum of divisors of n that are not of the form 3k+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A374058 Expansion of Product_{k>=1} (1 - x^(3*k-2)) * (1 - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A326394 Expansion of Sum_{k>=1} x^k * (1 + x^(2*k)) / (1 - x^(3*k)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

A326394 Expansion of Sum_{k>=1} x^k * (1 + x^(2k)) / (1 - x^(3k)).