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.

A050457 a(n) = Sum_{ d divides n, d==1 mod 4} d - Sum_{ d divides n, d==3 mod 4} d.

Original entry on oeis.org

1, 1, -2, 1, 6, -2, -6, 1, 7, 6, -10, -2, 14, -6, -12, 1, 18, 7, -18, 6, 12, -10, -22, -2, 31, 14, -20, -6, 30, -12, -30, 1, 20, 18, -36, 7, 38, -18, -28, 6, 42, 12, -42, -10, 42, -22, -46, -2, 43, 31, -36, 14, 54, -20, -60, -6, 36, 30, -58, -12, 62, -30, -42, 1, 84, 20, -66, 18, 44, -36, -70
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Multiplicative because it is the Inverse Moebius transform of [1 0 -3 0 5 0 -7 ...], which is multiplicative. - Christian G. Bower, May 18 2005

Links

Crossrefs

Column k=1 of A322143.

Programs

  • Maple
    with(numtheory):
A050457 := proc(n)
    local count1, count3, d;
    count1 := 0:
    count3 := 0:
    for d in numtheory[divisors](n) do
        if d mod 4 = 1 then
            count1 := count1+d
        elif d mod 4 = 3 then
            count3 := count3+d
        fi:
    end do:
    count1-count3;
end proc: # Ridouane Oudra, Feb 02 2020
# second Maple program:
a:= n-> add(`if`(d::odd, d*(-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 03 2020
  • Mathematica
    Table[Sum[KroneckerSymbol[-4, d] d , {d, Divisors[n]}], {n,  71}] (* Indranil Ghosh, Mar 16 2017 *)
f[p_, e_] := If[Mod[p, 4] == 1, (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    {a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 1, p*=kronecker(-4,p); (p^(e+1)-1)/(p-1)))))} /* Michael Somos, May 29 2005 */
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-4, d)*d))} /* Michael Somos, May 29 2005 */

Formula

a(n) is multiplicative with a(p^e)=1 if p=2, a(p^e)=(p^(e+1)-1)/(p-1) if p == 1 (mod 4), a(p^e)=((-p)^(e+1)-1)/(-p-1) if p == 3 (mod 4). - Michael Somos, May 29 2005
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
a(n) = Im(Sum_{d|n} d*i^d). - Ridouane Oudra, Feb 02 2020
a(n) = Sum_{d|n} d*sin(d*Pi/2). - Ridouane Oudra, Feb 18 2023

A321828 a(n) = Sum_{d|n, d==1 mod 4} d^12 - Sum_{d|n, d==3 mod 4} d^12.

Original entry on oeis.org

1, 1, -531440, 1, 244140626, -531440, -13841287200, 1, 282429005041, 244140626, -3138428376720, -531440, 23298085122482, -13841287200, -129746094281440, 1, 582622237229762, 282429005041, -2213314919066160, 244140626, 7355813669568000
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=12 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^12 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^12)^(e+1)-1)/(p^12-1), ((-p^12)^(e+1)-1)/(-p^12-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321828(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^12), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^12*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^12)^(e+1)-1)/(p^12-1) if p == 1 (mod 4) and ((-p^12)^(e+1)-1)/(-p^12-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^12*sin(d*Pi/2). - Ridouane Oudra, Sep 08 2024

A321821 a(n) = Sum_{d|n, d==1 mod 4} d^5 - Sum_{d|n, d==3 mod 4} d^5.

Original entry on oeis.org

1, 1, -242, 1, 3126, -242, -16806, 1, 58807, 3126, -161050, -242, 371294, -16806, -756492, 1, 1419858, 58807, -2476098, 3126, 4067052, -161050, -6436342, -242, 9768751, 371294, -14290100, -16806, 20511150, -756492, -28629150, 1, 38974100
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=5 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^5 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
Table[With[{d=Divisors[n]},Total[Select[d,Mod[#,4]==1&]^5]-Total[Select[ d,Mod[ #,4]==3&]^5]],{n,40}] (* Harvey P. Dale, Dec 15 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^5)^(e+1)-1)/(p^5-1), ((-p^5)^(e+1)-1)/(-p^5-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321821(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^5), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^5)^(e+1)-1)/(p^5-1) if p == 1 (mod 4) and ((-p^5)^(e+1)-1)/(-p^5-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^5*sin(d*Pi/2). - Ridouane Oudra, Jun 23 2024

A321822 a(n) = Sum_{d|n, d==1 mod 4} d^6 - Sum_{d|n, d==3 mod 4} d^6.

Original entry on oeis.org

1, 1, -728, 1, 15626, -728, -117648, 1, 530713, 15626, -1771560, -728, 4826810, -117648, -11375728, 1, 24137570, 530713, -47045880, 15626, 85647744, -1771560, -148035888, -728, 244156251, 4826810, -386889776, -117648, 594823322, -11375728
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=6 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^6 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^6)^(e+1)-1)/(p^6-1), ((-p^6)^(e+1)-1)/(-p^6-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321822(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^6), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^6)^(e+1)-1)/(p^6-1) if p == 1 (mod 4) and ((-p^6)^(e+1)-1)/(-p^6-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^6*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024

A321823 a(n) = Sum_{d|n, d==1 mod 4} d^7 - Sum_{d|n, d==3 mod 4} d^7.

Original entry on oeis.org

1, 1, -2186, 1, 78126, -2186, -823542, 1, 4780783, 78126, -19487170, -2186, 62748518, -823542, -170783436, 1, 410338674, 4780783, -893871738, 78126, 1800262812, -19487170, -3404825446, -2186, 6103593751, 62748518, -10455572420, -823542
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=7 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^7 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^7)^(e+1)-1)/(p^7-1), ((-p^7)^(e+1)-1)/(-p^7-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321823(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^7), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^7)^(e+1)-1)/(p^7-1) if p == 1 (mod 4) and ((-p^7)^(e+1)-1)/(-p^7-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^7*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024

A321824 a(n) = Sum_{d|n, d==1 mod 4} d^8 - Sum_{d|n, d==3 mod 4} d^8.

Original entry on oeis.org

1, 1, -6560, 1, 390626, -6560, -5764800, 1, 43040161, 390626, -214358880, -6560, 815730722, -5764800, -2562506560, 1, 6975757442, 43040161, -16983563040, 390626, 37817088000, -214358880, -78310985280, -6560, 152588281251, 815730722
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=8 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^8 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^8)^(e+1)-1)/(p^8-1), ((-p^8)^(e+1)-1)/(-p^8-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321824(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^8), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^8*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^8)^(e+1)-1)/(p^8-1) if p == 1 (mod 4) and ((-p^8)^(e+1)-1)/(-p^8-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^8*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024

Extensions

Keyword mult from Ilya Gutkovskiy, Dec 06 2018

A321825 a(n) = Sum_{d|n, d==1 (mod 4)} d^9 - Sum_{d|n, d==3 (mod 4)} d^9.

Original entry on oeis.org

1, 1, -19682, 1, 1953126, -19682, -40353606, 1, 387400807, 1953126, -2357947690, -19682, 10604499374, -40353606, -38441425932, 1, 118587876498, 387400807, -322687697778, 1953126, 794239673292, -2357947690, -1801152661462, -19682, 3814699218751
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=9 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^9 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^9)^(e+1)-1)/(p^9-1), ((-p^9)^(e+1)-1)/(-p^9-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321825(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^9), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^9)^(e+1)-1)/(p^9-1) if p == 1 (mod 4) and ((-p^9)^(e+1)-1)/(-p^9-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^9*sin(d*Pi/2). - Ridouane Oudra, Aug 18 2024

A321826 a(n) = Sum_{d|n, d==1 mod 4} d^10 - Sum_{d|n, d==3 mod 4} d^10.

Original entry on oeis.org

1, 1, -59048, 1, 9765626, -59048, -282475248, 1, 3486725353, 9765626, -25937424600, -59048, 137858491850, -282475248, -576640684048, 1, 2015993900450, 3486725353, -6131066257800, 9765626, 16679598443904, -25937424600, -41426511213648, -59048, 95367441406251, 137858491850
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=10 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^10 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^10)^(e+1)-1)/(p^10-1), ((-p^10)^(e+1)-1)/(-p^10-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321826(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^10), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^10*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^10)^(e+1)-1)/(p^10-1) if p == 1 (mod 4) and ((-p^10)^(e+1)-1)/(-p^10-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^10*sin(d*Pi/2). - Ridouane Oudra, Sep 04 2024

A321827 a(n) = Sum_{d|n, d==1 (mod 4)} d^11 - Sum_{d|n, d==3 (mod 4)} d^11.

Original entry on oeis.org

1, 1, -177146, 1, 48828126, -177146, -1977326742, 1, 31380882463, 48828126, -285311670610, -177146, 1792160394038, -1977326742, -8649707208396, 1, 34271896307634, 31380882463, -116490258898218, 48828126, 350275523038332, -285311670610
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=11 of A322143.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A000265.

Programs

  • Mathematica
    s[n_, r_] := DivisorSum[n, #^11 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^11)^(e+1)-1)/(p^11-1), ((-p^11)^(e+1)-1)/(-p^11-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
  • PARI
    apply( A321828(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^11), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^11*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^11)^(e+1)-1)/(p^11-1) if p == 1 (mod 4) and ((-p^11)^(e+1)-1)/(-p^11-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^11*sin(d*Pi/2). - Ridouane Oudra, Sep 08 2024

A050459 a(n) = Sum_{d|n, d==1 mod 4} d^3 - Sum_{d|n, d==3 mod 4} d^3.

Original entry on oeis.org

1, 1, -26, 1, 126, -26, -342, 1, 703, 126, -1330, -26, 2198, -342, -3276, 1, 4914, 703, -6858, 126, 8892, -1330, -12166, -26, 15751, 2198, -18980, -342, 24390, -3276, -29790, 1, 34580, 4914, -43092, 703, 50654, -6858, -57148, 126, 68922, 8892, -79506, -1330
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 1999

Keywords

Comments

Multiplicative because it is the Inverse Möbius transform of [1 0 -3^3 0 5^3 0 -7^3 ...], which is multiplicative. - Christian G. Bower, May 18 2005

Links

Crossrefs

Column k=3 of A322143.
Glaisher's E_i (i=0..12): A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.
Cf. A050451, A050454.

Programs

  • Maple
    A050459 := proc(n) local a; a := 0 ; for d in numtheory[divisors](n) do if d mod 4 = 1 then a := a+d^3 ; elif d mod 4 = 3 then a := a-d^3 ; end if; end do;  a ; end proc:
seq(A050459(n),n=1..100) ; # R. J. Mathar, Jan 07 2011
  • Mathematica
    s[n_, r_] := DivisorSum[n, #^3 &, Mod[#, 4]==r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Dec 06 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^3)^(e+1)-1)/(p^3-1), ((-p^3)^(e+1)-1)/(-p^3-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

Formula

a(n) = A050451(n) - A050454(n).
G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^3*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^3)^(e+1)-1)/(p^3-1) if p == 1 (mod 4) and ((-p^3)^(e+1)-1)/(-p^3-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^3*sin(d*Pi/2). - Ridouane Oudra, Jun 02 2024
Showing 1-10 of 10 results.

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