cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

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

Views

Author

Ralf Stephan, Apr 02 2003

Keywords

Comments

a(A004613(n))=0.

Links

Crossrefs

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

Programs

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

Formula

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

A293903 Sum of proper divisors of n of the form 4k+3.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 7, 3, 0, 0, 3, 0, 0, 10, 11, 0, 3, 0, 0, 3, 7, 0, 18, 0, 0, 14, 0, 7, 3, 0, 19, 3, 0, 0, 10, 0, 11, 18, 23, 0, 3, 7, 0, 3, 0, 0, 30, 11, 7, 22, 0, 0, 18, 0, 31, 10, 0, 0, 14, 0, 0, 26, 42, 0, 3, 0, 0, 18, 19, 18, 42, 0, 0, 30, 0, 0, 10, 0, 43, 3, 11, 0, 18, 7, 23, 34, 47, 19, 3, 0, 7, 14, 0, 0, 54, 0, 0, 60
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2017

Keywords

Links

Crossrefs

Cf. A050452, A091570, A293513, A293901.

Programs

  • Mathematica
    Array[DivisorSum[#, # &, Mod[#, 4] == 3 &] - Boole[Mod[#, 4] == 3] # &, 105] (* Michael De Vlieger, Oct 23 2017 *)
  • PARI
    A293903(n) = sumdiv(n,d,(d

Formula

a(n) = Sum_{d|n, d
a(n) = A091570(n) - A293901(n).
G.f.: Sum_{k>=1} (4*k-1) * x^(8*k-2) / (1 - x^(4*k-1)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 - 1/8 = 0.0806167... . - Amiram Eldar, Nov 27 2023

A363904 Expansion of Sum_{k>0} x^(3*k) / (1 - x^(4*k))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 0, 2, 5, 0, 0, 1, 5, 0, 3, 3, 6, 1, 0, 0, 8, 2, 0, 5, 8, 0, 4, 0, 11, 1, 0, 5, 11, 0, 0, 3, 11, 3, 5, 6, 12, 1, 2, 0, 14, 0, 0, 8, 17, 2, 6, 0, 15, 5, 0, 8, 19, 0, 0, 4, 17, 0, 7, 11, 18, 1, 0, 0, 24, 5, 5, 11, 20, 0, 8, 0, 21, 3, 0, 11, 23, 3, 0, 5, 25, 6
Offset: 1

Views

Author

Seiichi Manyama, Jun 27 2023

Keywords

Links

Crossrefs

Cf. A001842, A050452, A363903.
Cf. A113415, A363902.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # + 1 &, Mod[#, 4] == 3 &]/4; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (d%4==3)*(d+1))/4;

Formula

a(n) = (1/4) * Sum_{d|n, d==3 mod 4} (d+1) = (A001842(n) + A050452(n))/4.
G.f.: Sum_{k>0} k * x^(4*k-1) / (1 - x^(4*k-1)).

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

Views

Author

Ralf Stephan, Apr 02 2003

Keywords

Comments

a(A002145(n))=1.

Links

Crossrefs

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

Programs

  • Mathematica
    sd[n_]:= Total[Select[Divisors[n], !IntegerQ[(# - 3) / 4]&]]; Array[sd, 100] (* Vincenzo Librandi, May 17 2013 *)
  • PARI
    for(n=1,100,print1(sumdiv(n,d,if(d%4!=3,d))","))

Formula

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

A363392 Sum of divisors of 4*n-2 of form 4*k+3.

Original entry on oeis.org

0, 3, 0, 7, 3, 11, 0, 18, 0, 19, 10, 23, 0, 30, 0, 31, 14, 42, 0, 42, 0, 43, 18, 47, 7, 54, 0, 66, 22, 59, 0, 73, 0, 67, 26, 71, 0, 93, 18, 79, 30, 83, 0, 90, 0, 98, 34, 114, 0, 113, 0, 103, 60, 107, 0, 114, 0, 138, 42, 126, 11, 126, 0, 127, 46, 131, 26, 180, 0, 139, 50, 154, 0, 157, 0, 151, 54, 186, 0, 162, 30
Offset: 1

Views

Author

Seiichi Manyama, Jul 08 2023

Keywords

Crossrefs

Cf. A050452, A359290, A363259.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 2, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(4*n-2, d, (d%4==3)*d);

Formula

a(n) = A050452(4*n-2).
G.f.: Sum_{k>0} (4*k-1) * x^(2*k) / (1 - x^(4*k-1)).

A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 3, 2, 0, 4, 4, 2, 5, 6, 7, 8, 8, 12, 15, 12, 17, 26, 23, 24, 37, 40, 39, 50, 62, 66, 74, 86, 101, 116, 122, 144, 175, 184, 202, 246, 274, 294, 340, 388, 432, 480, 533, 610, 684, 742, 835, 956, 1045, 1144, 1299, 1450, 1586, 1758, 1965, 2182, 2400, 2638, 2941, 3268, 3560, 3922
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 25 2024

Keywords

Crossrefs

Cf. A000712, A022567, A035462, A050452, A261629, A374018.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Product[1/(1 - x^(4 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A050452(k) * a(n-k).
a(n) = Sum_{k=0..n} A035462(k) * A035462(n-k).
a(n) ~ Pi^(3/2) * exp(Pi*sqrt(n/3)) / (2*sqrt(3) * Gamma(1/4)^2 * n). - Vaclav Kotesovec, Jun 25 2024

A363359 Sum of divisors of 4*n-1 of form 4*k+3.

Original entry on oeis.org

3, 7, 11, 18, 19, 23, 30, 31, 42, 42, 43, 47, 54, 66, 59, 73, 67, 71, 93, 79, 83, 90, 98, 114, 113, 103, 107, 114, 138, 126, 126, 127, 131, 180, 139, 154, 157, 151, 186, 162, 163, 167, 193, 217, 179, 186, 198, 191, 252, 199, 210, 233, 211, 258, 222, 223, 227, 252, 282, 239, 273, 266, 251, 324, 266, 263, 270
Offset: 1

Views

Author

Seiichi Manyama, Jul 08 2023

Keywords

Crossrefs

Cf. A050452, A078703, A364085.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 1, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(4*n-1, d, (d%4==3)*d);

Formula

a(n) = A050452(4*n-1).
G.f.: Sum_{k>0} (4*k-1) * x^k / (1 - x^(4*k-1)).

A363407 Sum of divisors of 4*n-3 of form 4*k+3.

Original entry on oeis.org

0, 0, 3, 0, 0, 10, 0, 0, 14, 0, 0, 18, 7, 0, 22, 0, 0, 26, 0, 18, 30, 0, 0, 34, 0, 0, 60, 0, 0, 42, 11, 0, 46, 26, 0, 50, 0, 0, 54, 0, 30, 84, 0, 0, 62, 0, 0, 100, 0, 0, 70, 0, 30, 74, 38, 0, 93, 0, 0, 82, 0, 42, 86, 34, 0, 90, 0, 0, 140, 0, 0, 132, 0, 0, 140, 50, 0, 106, 0, 0, 110, 0, 54, 114, 0, 42, 156
Offset: 1

Views

Author

Seiichi Manyama, Jul 08 2023

Keywords

Crossrefs

Cf. A050452, A359240, A364084.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 3, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(4*n-3, d, (d%4==3)*d);

Formula

a(n) = A050452(4*n-3).
G.f.: Sum_{k>0} (4*k-1) * x^(3*k) / (1 - x^(4*k-1)).
Previous Showing 11-18 of 18 results.

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