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 1-6 of 6 results.

A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 10, 12, 11, 12, 14, 14, 18, 16, 18, 20, 19, 24, 22, 22, 23, 24, 31, 28, 30, 28, 30, 36, 31, 32, 34, 36, 42, 40, 38, 38, 42, 48, 42, 44, 43, 44, 60, 46, 47, 48, 50, 62, 54, 56, 54, 60, 66, 56, 58, 60, 59, 72, 62, 62, 73, 64, 84, 68
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Not multiplicative: a(3)*a(7) <> a(21), for example.

Links

Crossrefs

Cf. A001826, A002131, A050449, A050464, A050469, A222183.
Cf. A050461, A050462, A050463.

Programs

  • Maple
    A050460 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050460(n),n=1..40) ; # R. J. Mathar, Dec 20 2011
  • Mathematica
    a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#&]; Array[a, 70] (* Jean-François Alcover, Dec 01 2015 *)
  • PARI
    a(n)=sumdiv(n,d,if(n/d%4==1,d)) \\ Charles R Greathouse IV, Dec 04 2013

Formula

G.f.: Sum_{n>0} n*x^n/(1-x^(4*n)). - Vladeta Jovovic, Nov 14 2002
G.f.: Sum_{k>0} x^(4*k-3) / (1 - x^(4*k-3))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050464(n).
a(n) = A050469(n) + A050464(n).
a(n) = (A002131(n) + A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A222183. (End)

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

A285895 Sum of divisors d of n such that n/d is not congruent to 0 mod 4.

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 13, 18, 12, 24, 14, 24, 24, 24, 18, 39, 20, 36, 32, 36, 24, 48, 31, 42, 40, 48, 30, 72, 32, 48, 48, 54, 48, 78, 38, 60, 56, 72, 42, 96, 44, 72, 78, 72, 48, 96, 57, 93, 72, 84, 54, 120, 72, 96, 80, 90, 60, 144, 62, 96, 104, 96, 84, 144, 68
Offset: 1

Views

Author

Seiichi Manyama, Apr 28 2017

Keywords

Examples

			The divisors of 8 are 1, 2, 4, and 8. 8/1 == 0 (mod 4) and 8/2 == 0 (mod 4). Hence, a(8) = 4 + 8 = 12.

Links

Crossrefs

Cf. A002131 (k=2), A078708 (k=3), this sequence (k=4), A285896 (k=5).
Cf. A002131, A050460, A050464.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, 3*2^(e-1), (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)
  • PARI
    a(n)=sumdiv(n, d, if(n/d%4, d, 0)); \\ Andrew Howroyd, Jul 20 2018

Formula

G.f.: Sum_{k>=1} k*x^k*(1 + x^k + x^(2*k))/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 12 2019
a(n) = A050460(n) + A002131(n/2) + A050464(n), where A002131(.)=0 for non-integer argument. - R. J. Mathar, May 25 2020
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(2^e) = 3*2^(e-1) and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 5*Pi^2/64 = 0.7710628... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/4^s). - Amiram Eldar, Dec 30 2022

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

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

Views

Author

Ilya Gutkovskiy, Sep 11 2019

Keywords

Links

Crossrefs

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.
Cf. A050464, A363900.

Programs

  • Mathematica
    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}]

Formula

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

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

A050467 a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

Original entry on oeis.org

0, 0, 1, 0, 0, 16, 1, 0, 81, 0, 1, 256, 0, 16, 626, 0, 0, 1296, 1, 0, 2482, 16, 1, 4096, 0, 0, 6562, 256, 0, 10016, 1, 0, 14722, 0, 626, 20736, 0, 16, 28562, 0, 0, 39712, 1, 256, 50706, 16, 1, 65536, 2401, 0, 83522, 0, 0, 104992, 626, 4096, 130402
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A013663, A050463, A050468, A285989.
Cf. A050464, A050465, A050466.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n],Mod[n/#,4]==3&]^4],{n,60}] (* Harvey P. Dale, Jun 10 2023 *)
a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^4); \\ Amiram Eldar, Nov 05 2023

Formula

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050463(n).
a(n) = A050463(n) - A050468(n).
a(n) = (A285989(n) - A050468(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 31*zeta(5)/64 - 5*Pi^5/3072 = 0.00418296735902... . (End)

Extensions

Offset corrected by Amiram Eldar, Nov 05 2023
Showing 1-6 of 6 results.

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

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