A001817 -id:A001817 - cp's OEIS frontend

A363901 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^2.

1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 1, 3, 6, 4, 1, 9, 1, 1, 8, 7, 4, 9, 1, 3, 10, 6, 1, 16, 1, 5, 12, 9, 1, 13, 4, 3, 14, 8, 6, 21, 1, 4, 16, 11, 1, 17, 1, 9, 21, 14, 1, 26, 1, 1, 20, 16, 8, 21, 1, 7, 22, 12, 4, 31, 6, 9, 24, 15, 1, 32, 1, 3, 26, 14, 10, 36, 4, 6, 28, 27, 1, 29, 1, 16, 30, 16, 1, 41, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A001817, A078181.

Cf. A113415, A363903.

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

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

a(n) = (1/3) * Sum_{d|n, d==1 mod 3} (d+2) = (2 * A001817(n) + A078181(n))/3.

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

A359212 Number of divisors of 3n-2 of form 3k+1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 3, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 2, 4, 4, 2, 2, 5, 2, 2, 2, 6, 2, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 6, 2, 2, 3, 4, 4, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 6, 2, 2, 4, 6, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 5, 4, 2, 2, 4

Offset: 1

Seiichi Manyama, Dec 21 2022

Cf. A001817, A359211.

a[n_] := DivisorSum[3*n-2, 1 &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

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

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(3*k-2))))

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

A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).

Original entry on oeis.org

1, -1, 1, 0, 1, -1, 2, -2, 1, 0, 1, 0, 2, -2, 1, -1, 1, -1, 2, -1, 2, 0, 1, -2, 2, -2, 1, 0, 1, 0, 2, -3, 1, 0, 2, 0, 2, -2, 2, -2, 1, -2, 2, -1, 1, 0, 1, -1, 3, -1, 1, 0, 1, -1, 2, -4, 2, 0, 1, -1, 2, -2, 2, -2, 2, 0, 2, -1, 1, 0, 1, -2, 2, -2, 2, 0, 2, -2, 2, -3, 1, 0, 1, 0, 2, -2, 1, -2, 1, 0, 4, -1

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

sign

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

Cf. A001817, A048272, A364011, A364205, A364232.

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

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

A218442 a(n) = Sum_{k=0..n} floor(n/(3*k + 1)).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 11, 12, 14, 15, 17, 19, 21, 22, 25, 26, 27, 29, 32, 34, 36, 37, 39, 41, 43, 44, 48, 49, 51, 53, 56, 57, 59, 61, 63, 65, 67, 69, 73, 74, 76, 78, 81, 82, 84, 85, 88, 91, 94, 95, 99, 100, 101, 103, 107, 109, 111, 112, 115, 117, 119, 121, 125, 127, 129, 131, 134, 135, 139, 140, 142, 144, 146, 148, 152

Offset: 0

Benoit Cloitre, Oct 28 2012

Amiram Eldar, 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 A001817.

Cf. A001620, A218443, A256425.

d[n_] := DivisorSum[n, 1 &, Mod[#, 3] == 1 &]; d[0] = 0; Accumulate@Array[d, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

A218442[n]:=sum(floor(n/(3*k+1)),k,0,n)$
makelist(A218442[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n\3,(n\(3*k+1)))
```

a(n) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,3) - (1 - gamma)/3 = A256425 - (1 - A001620)/3 = 0.536879... (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

A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-2))^(-1/(3k-2)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 7020, 62460, 562140, 5984280, 67252680, 863165160, 11700148680, 173098134000, 2625661170000, 45310413258000, 782198417206800, 14310269286746400, 280333959468789600, 6002139207488767200, 129820528515538159200, 2934651197018947982400

Offset: 0

Seiichi Manyama, Jul 07 2023

nonn

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

Cf. A001817, A035382, A206303, A262947, A362697.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1,N, (1-x^(3*k-2))^(1/(3*k-2)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001817(k) * a(n-k)/(n-k)!.

A364582 a(n) is the least number with exactly n divisors of the form 3*k+1.

Original entry on oeis.org

1, 4, 16, 28, 80, 112, 320, 280, 784, 560, 1600, 1120, 10000, 2240, 3920, 2800, 25600, 5600, 1310720, 6160, 15680, 11200, 48400, 12320, 110000, 70000, 39200, 24640, 6553600, 30800, 5368709120, 36400, 78400, 179200, 440000, 61600, 343597383680, 1210000, 490000, 80080

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 1..200

Cf. A001817, A005179, A038547, A364583.

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

More terms from Bert Dobbelaere, Jul 31 2023

A363970 Expansion of Sum_{k>0} k^2 * x^(3k-2) / (1 - x^(3k-2)).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 10, 5, 1, 17, 1, 5, 26, 10, 1, 41, 1, 1, 50, 21, 10, 65, 1, 5, 82, 26, 1, 114, 1, 17, 122, 41, 1, 145, 10, 5, 170, 50, 26, 217, 1, 10, 226, 69, 1, 257, 1, 41, 299, 98, 1, 354, 1, 1, 362, 114, 50, 401, 1, 21, 442, 122, 10, 525, 26, 65, 530, 149, 1, 602, 1, 5, 626, 170

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001817, A363901.

Cf. A103637, A363975.

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

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

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

A363975 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^3.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 7, 4, 1, 11, 1, 4, 16, 7, 1, 25, 1, 1, 29, 14, 7, 37, 1, 4, 46, 16, 1, 65, 1, 11, 67, 25, 1, 79, 7, 4, 92, 29, 16, 119, 1, 7, 121, 40, 1, 137, 1, 25, 160, 56, 1, 190, 1, 1, 191, 65, 29, 211, 1, 14, 232, 67, 7, 278, 16, 37, 277, 82, 1, 317, 1, 4, 326, 92, 46, 383, 7, 16, 379

Offset: 1

Seiichi Manyama, Jun 30 2023

nonn

Cf. A001817, A363901.

Cf. A363970.

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

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

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

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

cp's OEIS Frontend

A363901 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^2.

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A359212 Number of divisors of 3*n-2 of form 3*k+1.

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PARI

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A364204 Expansion of Sum_{k>=0} x^(3*k+1) / (1 + x^(3*k+1)).

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Mathematica

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A218442 a(n) = Sum_{k=0..n} floor(n/(3*k + 1)).

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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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Maple

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A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-2))^(-1/(3*k-2)).

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Formula

A364582 a(n) is the least number with exactly n divisors of the form 3*k+1.

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PARI

A359212 Number of divisors of 3n-2 of form 3k+1.

A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).

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)).

A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-2))^(-1/(3k-2)).

A363970 Expansion of Sum_{k>0} k^2 * x^(3k-2) / (1 - x^(3k-2)).