A326399 -id:A326399 - cp's OEIS frontend

A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

1, 3, 3, 7, 6, 9, 8, 15, 9, 18, 12, 21, 14, 24, 18, 31, 18, 27, 20, 42, 24, 36, 24, 45, 31, 42, 27, 56, 30, 54, 32, 63, 36, 54, 48, 63, 38, 60, 42, 90, 42, 72, 44, 84, 54, 72, 48, 93, 57, 93, 54, 98, 54, 81, 72, 120, 60, 90, 60, 126, 62, 96, 72, 127, 84, 108, 68, 126, 72, 144

Offset: 1

Vladeta Jovovic, Dec 18 2002

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

Cf. A000203, A035191, A046913, A346933.
Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).
Cf. A326399, A326400.

f[p_, e_] := If[p == 3, 3^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

for(n=1,70,d=divisors(n); s=0; for(j=1,matsize(d)[2],if((n/d[j])%3>0,s=s+d[j])); print1(s,","))

PARI
```
a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)
```

G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.
a(n) = (A000203(3*n)-A000203(n))/3. - Vladeta Jovovic, Dec 22 2003
G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - Ilya Gutkovskiy, Sep 13 2019
From R. J. Mathar, May 25 2020: (Start)
a(n) = A326399(n) + A326400(n).
a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - Amiram Eldar, Dec 30 2022

Extended by Klaus Brockhaus and Benoit Cloitre, Dec 20 2002

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 5, 0, 7, 1, 6, 0, 8, 3, 10, 1, 9, 0, 15, 0, 13, 1, 15, 5, 14, 0, 16, 1, 21, 0, 21, 3, 19, 8, 18, 0, 20, 0, 35, 1, 24, 0, 27, 9, 25, 1, 30, 0, 36, 3, 28, 1, 27, 16, 40, 0, 31, 1, 45, 0, 32, 0, 42, 14, 39, 0, 39, 3, 56, 1, 45, 0, 38, 15, 40, 8, 42, 0, 71

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

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

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.

Cf. A050464, A363900.

nmax = 80; CoefficientList[Series[Sum[k x^(2 k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

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

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

A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

Original entry on oeis.org

1, 1, 3, 3, 4, 3, 8, 5, 9, 4, 10, 9, 14, 8, 12, 11, 16, 9, 20, 12, 24, 10, 22, 15, 21, 14, 27, 24, 28, 12, 32, 21, 30, 16, 32, 27, 38, 20, 42, 20, 40, 24, 44, 30, 36, 22, 46, 33, 57, 21, 48, 42, 52, 27, 40, 40, 60, 28, 58, 36, 62, 32, 72, 43, 56, 30, 68, 48, 66, 32

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia Rella, Resurgence, Stokes constants, and arithmetic functions in topological string theory, arXiv:2212.10606 [hep-th], 2022. See pages 21 - 23.

Cf. A000593, A002324, A050469, A078708, A326399, A326400.

Cf. A175646, A340577.

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 70}]
f[p_, e_] := Which[p == 3, p^e, Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 3] == 2, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 3, 3^f[i,2], if(f[i,1]%3 == 1, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1), (f[i,1]^(f[i,2]+1) + (-1)^f[i,2])/(f[i,1] + 1))));} \\ Amiram Eldar, Nov 06 2022

a(n) = Sum_{d|n, n/d==1 (mod 3)} d - Sum_{d|n, n/d==2 (mod 3)} d.
a(n) = A326399(n) - A326400(n).
Multiplicative with a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1 (mod 3), and (p^(e+1) + (-1)^e)/(p + 1) if p == 2 (mod 3). - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) * Product_{primes p == 2 (mod 3)} 1/(1 + 1/p^2) = (1/2) * A175646 * (2*Pi^2/27)/A340577 = 0.3906512064... . - Amiram Eldar, Nov 06 2022

A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 12, 14, 13, 14, 15, 17, 17, 21, 19, 20, 22, 24, 23, 28, 25, 27, 27, 28, 29, 35, 32, 34, 36, 34, 35, 43, 37, 38, 39, 40, 42, 51, 43, 48, 45, 47, 47, 59, 49, 50, 52, 54, 53, 63, 60, 57, 57, 58, 59, 70, 62, 64, 66, 68, 65, 84, 67, 68, 69, 70, 72, 86, 73, 74, 75, 77, 84, 94

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A363898, A363899, A363900.
Cf. A002131, A050460, A326399.
Cf. A001876, A284097.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d%5==1)*d);
```

a(n) = Sum_{d|n, n/d==1 mod 5} d.

G.f.: Sum_{k>0} x^(5*k-4) / (1 - x^(5*k-4))^2.

A364232 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1))^2.

Original entry on oeis.org

1, -2, 3, -3, 5, -6, 8, -10, 9, -9, 11, -9, 14, -16, 15, -19, 17, -18, 20, -17, 24, -21, 23, -30, 26, -28, 27, -24, 29, -27, 32, -42, 33, -33, 40, -27, 38, -40, 42, -53, 41, -48, 44, -35, 45, -45, 47, -57, 57, -47, 51, -42, 53, -54, 56, -80, 60, -57, 59, -51, 62, -64, 72, -83, 70, -63, 68, -53, 69, -72

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

sign

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

Cf. A002129, A186690, A326399, A364012, A364204, A364235.

nmax = 70; CoefficientList[Series[Sum[x^(3 k + 1)/(1 + x^(3 k + 1))^2, {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, MemberQ[{1}, Mod[n/#, 3]] &], {n, 1, 70}]

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

cp's OEIS Frontend

A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

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A326400 Expansion of Sum_{k>=1} k * x^(2*k) / (1 - x^(3*k)).

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A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

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A364232 Expansion of Sum_{k>=0} x^(3*k+1) / (1 + x^(3*k+1))^2.

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A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

A364232 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1))^2.