A001822 -id:A001822 - cp's OEIS frontend

A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

0, 1, 0, 1, 2, 1, 0, 4, 0, 3, 4, 1, 0, 6, 2, 4, 6, 1, 0, 10, 0, 5, 8, 4, 2, 10, 0, 6, 10, 3, 0, 15, 4, 7, 14, 1, 0, 14, 0, 13, 14, 6, 0, 20, 2, 9, 16, 4, 0, 20, 6, 10, 18, 1, 6, 28, 0, 11, 20, 10, 0, 22, 0, 15, 24, 5, 0, 30, 8, 20, 24, 4, 0, 26, 2, 14, 30, 10, 0, 40, 0, 15, 28, 6, 8

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A078182, A363901.

Cf. A113415, A363904.

a[n_] := DivisorSum[n, # + 1 &, Mod[#, 3] == 2 &]/3; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%3==2)*(d+1))/3;
```

a(n) = (1/3) * Sum_{d|n, d==2 mod 3} (d+1) = (A001822(n) + A078182(n))/3.

G.f.: Sum_{k>0} k * x^(3*k-1) / (1 - x^(3*k-1)).

A364205 Expansion of Sum_{k>=0} x^(3k+2) / (1 + x^(3k+2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A048272, A364014, A364204, A364235.

nmax = 92; CoefficientList[Series[Sum[x^(3 k + 2)/(1 + x^(3 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 92}]

a(n) = Sum_{d|n, n/d==2 (mod 3)} (-1)^(d+1).

A364583 a(n) is the least number with exactly n divisors of the form 3*k+2.

Original entry on oeis.org

1, 2, 8, 20, 40, 80, 140, 320, 280, 800, 560, 5120, 1120, 6400, 2240, 3920, 3080, 40000, 5600, 102400, 6160, 15680, 35840, 20971520, 12320, 110000, 44800, 39200, 24640, 1342177280, 30800, 193600, 40040, 250880, 280000, 440000, 61600, 1210000, 716800, 313600, 80080, 5497558138880

Offset: 0

Ilya Gutkovskiy, Jul 28 2023

nonn

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A001822, A005179, A038547, A364582.

a(n) = my(k=1); while (sumdiv(k, d, (d%3)==2) != n, k++); k; \\ Michel Marcus, Jul 29 2023

a(n) <= 5*2^(n-1). - David A. Corneth, Jul 29 2023

More terms from David A. Corneth, Jul 29 2023

A218443 a(n) = Sum_{k=0..n} floor(n/(3k+2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 4, 6, 6, 8, 9, 10, 10, 12, 13, 15, 16, 17, 17, 20, 20, 22, 23, 25, 26, 28, 28, 30, 31, 33, 33, 36, 37, 39, 41, 42, 42, 44, 44, 48, 49, 51, 51, 54, 55, 57, 58, 60, 60, 63, 64, 66, 67, 68, 70, 74, 74, 76, 77, 80, 80, 82, 82, 85, 87, 89, 89, 92, 93, 97, 98, 100, 100, 102, 103, 105, 107, 109, 109

Offset: 0

Benoit Cloitre, Oct 28 2012

Robert Israel, Table of n, a(n) for n = 0..10000
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.

Partial sums of A001822.

Cf. A001620, A218442, A256843.

N:= 100: # to get a(0)..a(N)
A001822:= Vector(N+1):
for m from 2 to N by 3 do
  L:= [seq(i,i=m+1..N+1,m)]:
  A001822[L]:= map(`+`,A001822[L],1)
od:
ListTools:-PartialSums(convert(A001822,list)); # Robert Israel, Feb 28 2017

Table[Sum[Floor[n/(3k+2)],{k,0,n}],{n,0,80}] (* Harvey P. Dale, Jun 22 2013 *)
d[n_] := DivisorSum[n, 1 &, Mod[#, 3] == 2 &]; d[0] = 0; Accumulate@Array[d, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

A218443[n]:=sum(floor(n/(3*k+2)),k,0,n)$
makelist(A218443[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n\3,(n\(3*k+2)))
```

G.f.: Sum_{k>=0} x^(3*k+2)/((1-x^(3*k+2))*(1-x)). - Robert Israel, Feb 28 2017

a(n) = 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

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

A326395 Expansion of Sum_{k>=1} x^(2k) (1 + x^k) / (1 - x^(3*k)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 0, 2, 2, 2, 1, 4, 0, 2, 3, 2, 1, 5, 0, 3, 2, 2, 1, 6, 1, 2, 3, 2, 1, 6, 0, 3, 3, 2, 2, 7, 0, 2, 2, 4, 1, 6, 0, 3, 5, 2, 1, 7, 0, 3, 3, 2, 1, 7, 2, 4, 2, 2, 1, 9, 0, 2, 4, 3, 2, 6, 0, 3, 3, 4, 1, 10, 0, 2, 4, 2, 2, 6, 0, 5, 4, 2, 1, 8, 2, 2, 3, 4, 1, 10

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Number of divisors of n that are not of the form 3*k + 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001817, A001822, A004611 (positions of 0's), A007494, A035191, A082050, A326394.

Cf. A001620, A256425.

N:= 100: # for a(1) .. a(N)
S:= series(add(x^(2*k)*(1+x^k)/(1-x^(3*k)),k=1..N/2),x,N+1):
seq(coeff(S,x,i),i=1..N); # Robert Israel, Aug 27 2020

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

a(n) = {numdiv(n) - sumdiv(n, d, d%3==1)} \\ Andrew Howroyd, Sep 11 2019

a(n) = A000005(n) - A001817(n).

Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(1,3) = (5*A001620-2)/3 - A256425 = -0.382447... . - Amiram Eldar, Jan 14 2024

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 9, 24, 225, 504, 16065, 27216, 1555281, 6123600, 159249321, 779262120, 31816914129, 240363179784, 8207359913025, 66059979227424, 2145292484152545, 19782668403572256, 1015331126023222281, 7961977144683689400, 454920488042137314561

Offset: 0

Seiichi Manyama, Jul 07 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A001822, A035386, A206303, A262946, A362696.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^(3*k-1))^(1/(3*k-1)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001822(k) * a(n-k)/(n-k)!.

A364357 Number of divisors of n of the form 3*k+2 that are at most sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 21 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001822, A038548, A364209.

N:= 100: # for a(1) .. a(N)
M:= floor((sqrt(N)-3)/2):
G:= series(add(x^((3*k+2)^2)/(1-x^(3*k+2)),k=0..M),x,N+1):
seq(coeff(G,x,i),i=1..N); # Robert Israel, Jun 05 2024

Table[Count[Divisors[n], _?(# <= Sqrt[n] && MemberQ[{2}, Mod[#, 3]] &)], {n, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(3 k + 2)^2/(1 - x^(3 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=0} x^((3*k+2)^2) / (1 - x^(3*k+2)).

A363971 Expansion of Sum_{k>0} k^2 * x^(3k-1) / (1 - x^(3k-1)).

Original entry on oeis.org

0, 1, 0, 1, 4, 1, 0, 10, 0, 5, 16, 1, 0, 26, 4, 10, 36, 1, 0, 54, 0, 17, 64, 10, 4, 82, 0, 26, 100, 5, 0, 131, 16, 37, 148, 1, 0, 170, 0, 63, 196, 26, 0, 242, 4, 65, 256, 10, 0, 294, 36, 82, 324, 1, 20, 396, 0, 101, 400, 54, 0, 442, 0, 131, 488, 17, 0, 566, 64, 174, 576, 10, 0, 626, 4, 170

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001822, A363902.

Cf. A103638, A363976.

a[n_] := DivisorSum[n, ((#+1)/3)^2 &, Mod[#, 3] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%3==2)*((d+1)/3)^2);

a(n) = Sum_{d|n, d==2 mod 3} ((d+1)/3)^2.

cp's OEIS Frontend

A363902 Expansion of Sum_{k>0} x^(2*k) / (1 - x^(3*k))^2.

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A364205 Expansion of Sum_{k>=0} x^(3*k+2) / (1 + x^(3*k+2)).

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A364583 a(n) is the least number with exactly n divisors of the form 3*k+2.

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A218443 a(n) = Sum_{k=0..n} floor(n/(3k+2)).

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Maple

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Maxima

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

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A326395 Expansion of Sum_{k>=1} x^(2*k) * (1 + x^k) / (1 - x^(3*k)).

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Programs

Maple

Mathematica

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A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-1))^(-1/(3*k-1)).

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A364357 Number of divisors of n of the form 3*k+2 that are at most sqrt(n).

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Author

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A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

A364205 Expansion of Sum_{k>=0} x^(3k+2) / (1 + x^(3k+2)).

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

A326395 Expansion of Sum_{k>=1} x^(2k) (1 + x^k) / (1 - x^(3*k)).

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).

A363971 Expansion of Sum_{k>0} k^2 * x^(3k-1) / (1 - x^(3k-1)).