A078181 -id:A078181 - cp's OEIS frontend

A363514 Sum of divisors of 3n-1 of form 3k+1.

1, 1, 5, 1, 8, 1, 15, 1, 14, 1, 21, 8, 20, 1, 27, 1, 36, 1, 40, 1, 32, 14, 39, 1, 38, 8, 71, 1, 44, 1, 51, 20, 57, 1, 70, 1, 88, 1, 63, 8, 62, 26, 85, 1, 68, 1, 120, 14, 74, 1, 100, 32, 80, 8, 87, 1, 130, 1, 131, 1, 112, 38, 99, 1, 98, 1, 180, 8, 104, 20, 111, 44, 110, 14, 168, 1, 172, 1

Offset: 1

Seiichi Manyama, Jun 26 2023

nonn

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

Cf. A363889, A363890, A363891.

Cf. A078181, A359211.

a[n_] := DivisorSum[3*n - 1, # &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 26 2023 *)

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

a(n) = A078181(3*n-1).

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

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

Original entry on oeis.org

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)).

A363889 Sum of divisors of 3n-2 of form 3k+1.

Original entry on oeis.org

1, 5, 8, 11, 14, 21, 20, 23, 26, 40, 32, 35, 38, 55, 44, 47, 57, 70, 56, 59, 62, 85, 68, 88, 74, 100, 80, 83, 86, 115, 112, 95, 98, 140, 104, 107, 110, 168, 116, 119, 122, 160, 128, 154, 160, 175, 140, 143, 146, 190, 152, 184, 158, 231, 164, 167, 183, 220, 208, 179, 182, 235, 188

Offset: 1

Seiichi Manyama, Jun 26 2023

nonn

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

Cf. A363514, A363890, A363891.

Cf. A078181, A359212.

a[n_] := DivisorSum[3*n - 2, # &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 26 2023 *)

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

a(n) = A078181(3*n-2).

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

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

Original entry on oeis.org

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

A082051 Sum of divisors of n that are not of the form 3k+2.

Original entry on oeis.org

1, 1, 4, 5, 1, 10, 8, 5, 13, 11, 1, 26, 14, 8, 19, 21, 1, 37, 20, 15, 32, 23, 1, 50, 26, 14, 40, 40, 1, 65, 32, 21, 37, 35, 8, 89, 38, 20, 56, 55, 1, 80, 44, 27, 73, 47, 1, 114, 57, 36, 55, 70, 1, 118, 56, 40, 80, 59, 1, 141, 62, 32, 104, 85, 14, 131, 68, 39, 73, 88, 1, 185

Offset: 1

Ralf Stephan, Apr 02 2003

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

Cf. A000203, A003627, A027748, A046913, A078181, A078182, A082050, A086463, A326394.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 2) / 3]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)

for(n=1,100,print1(sumdiv(n,d,if(d%3!=2,d))","))

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, (3*n-2)*x^(3*n-2)/(1-x^(3*n-2)) + (3*n)*x^(3*n)/(1-x^(3*n)) );
v = Vec(gf)
\\ Joerg Arndt, May 17 2013

a(A003627(n)) = 1.
G.f.: Sum_{k>=1} x^k*(1 + 3*x^(2*k) + 2*x^(3*k))/(1 - x^(3*k))^2. - Ilya Gutkovskiy, Sep 12 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Jan 06 2024

A082052 Sum of divisors of n that are not of the form 4k+1.

Original entry on oeis.org

0, 2, 3, 6, 0, 11, 7, 14, 3, 12, 11, 27, 0, 23, 18, 30, 0, 29, 19, 36, 10, 35, 23, 59, 0, 28, 30, 55, 0, 66, 31, 62, 14, 36, 42, 81, 0, 59, 42, 84, 0, 74, 43, 83, 18, 71, 47, 123, 7, 62, 54, 84, 0, 110, 66, 119, 22, 60, 59, 162, 0, 95, 73, 126, 0, 110, 67, 108, 26, 138, 71

Offset: 1

Ralf Stephan, Apr 02 2003

a(A004613(n))=0.

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

Cf. A000203, A050449, A050452, A050460, A078181, A078182, A082053.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 1) / 4]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)
Table[DivisorSum[n,#&,(!IntegerQ[(#-1)/4]&)],{n,80}] (* Harvey P. Dale, Nov 30 2019 *)

for(n=1,100,print1(sumdiv(n,d,if(d%4!=1,d))","))

G.f.: Sum_{k>=1} x^(2*k)*(2 + 3*x^k + 4*x^(2*k) + 2*x^(4*k) + x^(5*k))/(1 - x^(4*k))^2. - Ilya Gutkovskiy, Sep 12 2019

A082053 Sum of divisors of n that are not of the form 4k+3.

Original entry on oeis.org

1, 3, 1, 7, 6, 9, 1, 15, 10, 18, 1, 25, 14, 17, 6, 31, 18, 36, 1, 42, 22, 25, 1, 57, 31, 42, 10, 49, 30, 54, 1, 63, 34, 54, 6, 88, 38, 41, 14, 90, 42, 86, 1, 73, 60, 49, 1, 121, 50, 93, 18, 98, 54, 90, 6, 113, 58, 90, 1, 150, 62, 65, 31, 127, 84, 130, 1, 126, 70, 102, 1, 192

Offset: 1

Ralf Stephan, Apr 02 2003

a(A002145(n))=1.

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

Cf. A000203, A050449, A050452, A050460, A078181, A078182, A082052.

sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 3) / 4]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)

for(n=1,100,print1(sumdiv(n,d,if(d%4!=3,d))","))

G.f.: Sum_{k>=1} x^k*(1 + 2*x^k + 4*x^(3*k) + 3*x^(4*k) + 2*x^(5*k))/(1 - x^(4*k))^2. - Ilya Gutkovskiy, Sep 12 2019

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 46, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 68, 35, 24, 1, 1, 13, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 13, 1

Offset: 1

Seiichi Manyama, Jan 17 2023

nonn

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

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8), this sequence (k=11).

Cf. A357911.

a[n_] := DivisorSum[n, # &, Mod[#, 11] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

a(n) = sumdiv(n, d, (Mod(d, 11)==1)*d);

my(N=100, x='x+O('x^N)); Vec(sum(k=0, N, (11*k+1)*x^(11*k+1)/(1-x^(11*k+1))))

G.f.: Sum_{k>=0} (11*k+1) * x^(11*k+1)/(1 - x^(11*k+1)).

A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2k)) / (1 + x^k + x^(2k))^2.

Original entry on oeis.org

1, -1, 1, 3, -4, -1, 8, -5, 1, 4, -10, 3, 14, -8, -4, 11, -16, -1, 20, -12, 8, 10, -22, -5, 21, -14, 1, 24, -28, 4, 32, -21, -10, 16, -32, 3, 38, -20, 14, 20, -40, -8, 44, -30, -4, 22, -46, 11, 57, -21, -16, 42, -52, -1, 40, -40, 20, 28, -58, -12, 62, -32, 8, 43, -56, 10

Offset: 1

Ilya Gutkovskiy, Sep 12 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 33 - 35.

Cf. A002129, A050457, A078181, A078182, A078708, A162397 (Moebius transform), A326401.

nmax = 66; CoefficientList[Series[Sum[x^k (1 - x^(2 k))/(1 + x^k + x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[#, 3]] &], {n, 1, 66}]
f[p_, e_] := If[Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), ((-p)^(e + 1) - 1)/(-p - 1)]; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 28 2023 *)

a(n)={sumdiv(n, d, d*((d+1)%3-1))} \\ Andrew Howroyd, Sep 12 2019

a(n) = Sum_{d|n, d==1 (mod 3)} d - Sum_{d|n, d==2 (mod 3)} d.
a(n) = A078181(n) - A078182(n).
Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1 (mod 3) and a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2 (mod 3). - Amiram Eldar, Nov 28 2023

cp's OEIS Frontend

A363514 Sum of divisors of 3*n-1 of form 3*k+1.

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

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A363889 Sum of divisors of 3*n-2 of form 3*k+1.

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

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A082051 Sum of divisors of n that are not of the form 3k+2.

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PARI

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A082052 Sum of divisors of n that are not of the form 4k+1.

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Mathematica

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A082053 Sum of divisors of n that are not of the form 4k+3.

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Mathematica

PARI

Formula

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

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A363514 Sum of divisors of 3n-1 of form 3k+1.

A363889 Sum of divisors of 3n-2 of form 3k+1.

A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2k)) / (1 + x^k + x^(2k))^2.