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.

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)

A050461 a(n) = Sum_{d|n, n/d=1 mod 4} d^2.

Original entry on oeis.org

1, 4, 9, 16, 26, 36, 49, 64, 82, 104, 121, 144, 170, 196, 234, 256, 290, 328, 361, 416, 442, 484, 529, 576, 651, 680, 738, 784, 842, 936, 961, 1024, 1090, 1160, 1274, 1312, 1370, 1444, 1530, 1664, 1682, 1768, 1849, 1936, 2132, 2116, 2209
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Not multiplicative: a(3)*a(7) <> a(21), for example. - R. J. Mathar, Dec 20 2011

Links

Crossrefs

Cf. A050465, A050470, A076577, A027750, A002117.
Cf. A050460, A050462, A050463.

Programs

  • Haskell
    a050461 n = sum [d ^ 2 | d <- a027750_row n, mod (div n d) 4 == 1]
-- Reinhard Zumkeller, Mar 06 2012
  • Maple
    A050461 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d^2 ;
                end if;
        end do:
        a;
end proc:
seq(A050461(n),n=1..40) ; # R. J. Mathar, Dec 20 2011
  • Mathematica
    a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#^2&]; Array[a, 50] (* Jean-François Alcover, Feb 12 2018 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d % 4 == 1) * d^2); \\ Amiram Eldar, Nov 05 2023

Formula

a(n) = A050470(n) + A050465(n). - Reinhard Zumkeller, Mar 06 2012
From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A076577(n) - A050465(n).
a(n) = (A050470(n) + A076577(n))/2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/64 + 7*zeta(3)/16 = 1.010372968262... . (End)

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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