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.

Showing 1-4 of 4 results.

A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 4, 0, 2, 6, 0, 0, 6, 1, 0, 10, 2, 1, 8, 0, 0, 10, 4, 0, 12, 1, 0, 14, 0, 6, 12, 0, 2, 14, 0, 0, 20, 1, 4, 18, 2, 1, 16, 7, 0, 18, 0, 0, 20, 6, 8, 22, 0, 1, 24, 0, 2, 31, 0, 0, 28, 1, 0, 26, 12, 1, 24, 0, 0, 31, 4, 18, 28, 1, 0, 30, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A002131, A050460, A050469, A247037, A326400, A363900.
Cf. A050465, A050466, A050467.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)
  • PARI
    a(n)={sumdiv(n, d, d*(n/d%4==3))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} k*x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 13 2019
G.f.: Sum_{k>0} x^(4*k-1) / (1 - x^(4*k-1))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050460(n).
a(n) = A050460(n) - A050469(n).
a(n) = (A002131(n) - A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A247037. (End)

Extensions

Offset changed to 1 by Ilya Gutkovskiy, Sep 13 2019

A050465 a(n) = Sum_{d|n, n/d=3 mod 4} d^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 1, 0, 9, 0, 1, 16, 0, 4, 26, 0, 0, 36, 1, 0, 58, 4, 1, 64, 0, 0, 82, 16, 0, 104, 1, 0, 130, 0, 26, 144, 0, 4, 170, 0, 0, 232, 1, 16, 234, 4, 1, 256, 49, 0, 290, 0, 0, 328, 26, 64, 370, 0, 1, 416, 0, 4, 523, 0, 0, 520, 1, 0, 538, 104, 1, 576, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A002117, A027750, A050461, A050470, A076577.
Cf. A050464, A050466, A050467.

Programs

  • Haskell
    a050465 n = sum [d ^ 2 | d <- a027750_row n, mod (div n d) 4 == 3]
-- Reinhard Zumkeller, Mar 06 2012
  • Mathematica
    a[n_] := DivisorSum[n, #^2 &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^2); \\ Amiram Eldar, Nov 05 2023

Formula

a(n) = A050461(n) - A050470(n). - Reinhard Zumkeller, Mar 06 2012
From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A076577(n) - A050461(n).
a(n) = (A076577(n) - A050470(n))/2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 7*zeta(3)/16 - Pi^3/64 = 0.041426822002... . (End)

Extensions

Offset fixed by Reinhard Zumkeller, Mar 06 2012

A050463 a(n) = Sum_{d|n, n/d=1 mod 4} d^4.

Original entry on oeis.org

1, 16, 81, 256, 626, 1296, 2401, 4096, 6562, 10016, 14641, 20736, 28562, 38416, 50706, 65536, 83522, 104992, 130321, 160256, 194482, 234256, 279841, 331776, 391251, 456992, 531522, 614656, 707282, 811296, 923521, 1048576, 1185922
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A050448, A050467, A050468, A013663, A285989.
Cf. A050460, A050461, A050462.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d % 4 == 1) * d^4); \\ Amiram Eldar, Nov 05 2023

Formula

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050467(n).
a(n) = A050468(n) + A050467(n).
a(n) = (A050468(n) + A285989(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/3072 + 31*zeta(5)/64 = 1.000340795436113... . (End)

Extensions

Offset changed from 0 to 1 by Seiichi Manyama, Jul 08 2023

A050466 a(n) = Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

0, 0, 1, 0, 0, 8, 1, 0, 27, 0, 1, 64, 0, 8, 126, 0, 0, 216, 1, 0, 370, 8, 1, 512, 0, 0, 730, 64, 0, 1008, 1, 0, 1358, 0, 126, 1728, 0, 8, 2198, 0, 0, 2960, 1, 64, 3402, 8, 1, 4096, 343, 0, 4914, 0, 0, 5840, 126, 512, 6886, 0, 1, 8064, 0, 8, 9991, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

From Robert G. Wilson v, Mar 26 2015: (Start)
a(n) = 0 for n = 1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, ... (A072437).
a(n) = 1 for n = 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ... (A002145). (End)

Links

Crossrefs

Cf. A007331, A050462, A050471, A175572.
Cf. A050464, A050465, A050467.
Cf. A002145, A072437.

Programs

  • Mathematica
    a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 3 &])^3]; Array[a, 64] (* Robert G. Wilson v, Mar 26 2015 *)
a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, ((n/d % 4)== 3)* d^3); \\ Michel Marcus, Mar 26 2015

Formula

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A007331(n) - A050462(n).
a(n) = A050462(n) - A050471(n).
a(n) = (A007331(n) - A050471(n))/2.
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = Pi^4/192 - A175572/2 = 0.0128667399315... . (End)

Extensions

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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