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 1-10 of 10 results.

A050449 a(n) = Sum_{d|n, d == 1 (mod 4)} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 10, 6, 1, 1, 14, 1, 6, 1, 18, 10, 1, 6, 22, 1, 1, 1, 31, 14, 10, 1, 30, 6, 1, 1, 34, 18, 6, 10, 38, 1, 14, 6, 42, 22, 1, 1, 60, 1, 1, 1, 50, 31, 18, 14, 54, 10, 6, 1, 58, 30, 1, 6, 62, 1, 31, 1, 84, 34, 1, 18, 70, 6, 1, 10, 74, 38, 31, 1
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. A000593, A050452, A050460, A001826, A035451, A245058.
Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), this sequence (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

Programs

  • Maple
    A050449 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if d mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050449(n),n=1..40) ; # R. J. Mathar, Dec 20 2011
  • Mathematica
    a[n_] := DivisorSum[n, Boole[Mod[#, 4] == 1]*#&]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2018 *)
  • PARI
    a(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ Michel Marcus, Jan 30 2018

Formula

G.f.: Sum_{n>=0} (4*n+1)*x^(4*n+1)/(1-x^(4*n+1)). - Vladeta Jovovic, Nov 14 2002
a(n) = A000593(n) - A050452(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{n >= 1} x^n*(1 + 3*x^(4*n))/(1 - x^(4*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 = 0.205616... (A245058). - Amiram Eldar, Nov 26 2023

Extensions

More terms from Vladeta Jovovic, Nov 14 2002
More terms from Reinhard Zumkeller, Apr 18 2006

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)

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

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

A326399 Expansion of Sum_{k>=1} k * x^k / (1 - x^(3*k)).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 8, 10, 9, 11, 11, 15, 14, 16, 15, 21, 17, 18, 20, 27, 24, 23, 23, 30, 26, 28, 27, 40, 29, 33, 32, 42, 33, 35, 40, 45, 38, 40, 42, 55, 41, 48, 44, 57, 45, 47, 47, 63, 57, 57, 51, 70, 53, 54, 56, 80, 60, 59, 59, 81, 62, 64, 72, 85, 70, 69, 68, 87, 69, 88
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 11 2019

Keywords

Links

Crossrefs

Cf. A001817, A002131, A016777, A050460, A078181, A078708, A326400, A326401.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[k x^k/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, n/d==1 (mod 3)} d.
G.f.: Sum_{k>0} x^(3*k-2) / (1 - x^(3*k-2))^2. - Seiichi Manyama, Jun 29 2023

A050462 a(n) = Sum_{d|n, n/d=1 mod 4} d^3.

Original entry on oeis.org

1, 8, 27, 64, 126, 216, 343, 512, 730, 1008, 1331, 1728, 2198, 2744, 3402, 4096, 4914, 5840, 6859, 8064, 9262, 10648, 12167, 13824, 15751, 17584, 19710, 21952, 24390, 27216, 29791, 32768, 35938, 39312, 43218, 46720, 50654, 54872, 59346
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A007331, A050466, A050471, A175572.
Cf. A050460, A050461, A050463.

Programs

  • Mathematica
    a[n_] := Total[(n/Select[Divisors@ n, Mod[#, 4] == 1 &])^3]; Array[a, 39] (* Robert G. Wilson v, Mar 26 2015 *)
a[n_] := DivisorSum[n, #^3 &, Mod[n/#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, ((n/d % 4)== 1)* d^3); \\ Michel Marcus, Mar 26 2015

Formula

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A007331(n) - A050466(n).
a(n) = A050471(n) + A050466(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 = 1.00181129167264... . (End)

Extensions

Offset changed from 0 to 1 by Robert G. Wilson v, Mar 27 2015

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

A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 12, 14, 13, 14, 15, 17, 17, 21, 19, 20, 22, 24, 23, 28, 25, 27, 27, 28, 29, 35, 32, 34, 36, 34, 35, 43, 37, 38, 39, 40, 42, 51, 43, 48, 45, 47, 47, 59, 49, 50, 52, 54, 53, 63, 60, 57, 57, 58, 59, 70, 62, 64, 66, 68, 65, 84, 67, 68, 69, 70, 72, 86, 73, 74, 75, 77, 84, 94
Offset: 1

Views

Author

Seiichi Manyama, Jun 27 2023

Keywords

Links

Crossrefs

Cf. A363898, A363899, A363900.
Cf. A002131, A050460, A326399.
Cf. A001876, A284097.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d%5==1)*d);

Formula

a(n) = Sum_{d|n, n/d==1 mod 5} d.
G.f.: Sum_{k>0} x^(5*k-4) / (1 - x^(5*k-4))^2.

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

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
Showing 1-10 of 10 results.

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

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