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 11 results. Next

A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

Original entry on oeis.org

1, 13, 73, 205, 601, 949, 2353, 3277, 5905, 7813, 14521, 14965, 28393, 30589, 43873, 52429, 83233, 76765, 129961, 123205, 171769, 188773, 279313, 239221, 375601, 369109, 478297, 482365, 706441, 570349, 922561, 838861, 1060033, 1082029
Offset: 1

Views

Author

Benoit Cloitre, Jun 21 2003

Keywords

Links

Crossrefs

Cf. A001157, A001159, A002117, A007434, A013663, A065827.
Cf. A068963, A372962, A372963, A373007.
Cf. A057660, A084220, A372966, A373105.

Programs

  • Maple
    with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n),n=1..40); # Muniru A Asiru, Oct 09 2018
  • Mathematica
    Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* G. C. Greubel, Oct 08 2018 *)
f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)
  • PARI
    a(n)=sumdiv(n^2,d,d^4)/sumdiv(n^2,d,d^2)
  • PARI
    a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ Michel Marcus, Oct 09 2018

Formula

Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - Amiram Eldar, Oct 30 2022
From Peter Bala, Jan 18 2024: (Start)
a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.
Dirichlet g.f.: zeta(s) * zeta(s-4) / zeta(s-2) [Corrected by Michael Shamos, May 18 2025]. (End)
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_4(d). - Seiichi Manyama, May 18 2024

A372966 a(n) = sigma_8(n^2)/sigma_4(n^2).

Original entry on oeis.org

1, 241, 6481, 61681, 390001, 1561921, 5762401, 15790321, 42521761, 93990241, 214344241, 399754561, 815702161, 1388738641, 2527596481, 4042322161, 6975673921, 10247744401, 16983432721, 24055651681, 37346120881, 51656962081, 78310705441, 102337070401
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Comments

Apparently, a(n) == 1 (mod 240). - Hugo Pfoertner, May 20 2024

Links

Crossrefs

Cf. A057660, A084218, A084220, A108223.
Cf. A008683, A013956.
Cf. A372965.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 20 2024 *)
  • PARI
    a(n) = sigma(n^2, 8)/sigma(n^2, 4);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 8));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^4.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_8(d).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e + 4) + 1)/(p^4 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-4).
Sum_{k=1..n} a(k) ~ (zeta(9)/(9*zeta(5))) * n^9. (End)

A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 121, 2161, 15481, 78001, 261481, 823201, 1981561, 4726081, 9438121, 19485841, 33454441, 62746321, 99607321, 168560161, 253639801, 410333761, 571855801, 893864881, 1207533481, 1778937361, 2357786761, 3404813281, 4282153321, 6093828001, 7592304841, 10335939121
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A068970, A084220, A372950.
Cf. A008683, A013955.
Cf. A013663, A013666.
Cf. A350156, A372962.

Programs

  • Mathematica
    f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.

Original entry on oeis.org

1, 61, 721, 3901, 15601, 43981, 117601, 249661, 525601, 951661, 1771441, 2812621, 4826641, 7173661, 11248321, 15978301, 24137281, 32061661, 47045521, 60859501, 84790321, 108057901, 148035361, 180005581, 243765601, 294425101, 383163121, 458761501, 594822481, 686147581
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A068963, A084218, A372962.
Cf. A084220.
Cf. A013663, A013665.

Programs

  • Mathematica
    f[p_, e_] := (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 6));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_6(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(5) = 0.972439277... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.

Original entry on oeis.org

1, 249, 6535, 63737, 390501, 1627215, 5764459, 16316665, 42876109, 97234749, 214357551, 416521295, 815728525, 1435350291, 2551924035, 4177066233, 6975752529, 10676151141, 16983556183, 24889362237, 37670739565, 53375030199, 78310973115, 106629405775
Offset: 1

Views

Author

Seiichi Manyama, May 24 2024

Keywords

Links

Crossrefs

Cf. A068970, A084220, A372950, A372964.
Cf. A372952, A372963.
Cf. A372966.
Cf. A013664, A013667.

Programs

  • Mathematica
    f[p_, e_] := (p^(8*e + 8) - p^(8*e + 3) + p^3 - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 24 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));
  • PARI
    a(n) = sumdiv(n,d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_8(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+8) - p^(8*e+3) + p^3 - 1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(9)/zeta(6) = 0.984926747... . (End)
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5. - Seiichi Manyama, May 25 2024

A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

Original entry on oeis.org

1, 993, 58807, 1016801, 9762501, 58395351, 282458443, 1041204193, 3472494301, 9694163493, 25937263551, 59795016407, 137858120557, 280481233899, 574103396307, 1066193093601, 2015992480593, 3448186840893, 6131063781703, 9926520779301
Offset: 1

Views

Author

Seiichi Manyama, May 25 2024

Keywords

Links

Crossrefs

Cf. A057660, A084218, A084220, A372966.
Cf. A371491, A371878, A373007, A373103.
Cf. A013664, A013669.

Programs

  • Mathematica
    f[p_, e_] := (p^(10*e+5) + 1)/(p^5 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, May 25 2024 *)
  • PARI
    a(n) = sigma(n^2, 10)/sigma(n^2, 5);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^5*sigma(d, 10));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x^5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^5 * sigma_10(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(10*e+5) + 1)/(p^5 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(s-5).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = zeta(11)/zeta(6) = 0.9834383562... . (End)

A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

Original entry on oeis.org

1, 106, 1041, 7218, 19345, 110346, 136801, 465522, 768327, 2050570, 1947121, 7513938, 5226481, 14500906, 20138145, 29822066, 25640641, 81442662, 49651921, 139632210, 142409841, 206394826, 154751521, 484608402, 302749845, 554006986, 560366223, 987429618, 616040881
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A059376, A084220.
Cf. A373130, A373131.
Cf. A062952, A373132, A373135.
Cf. A013662, A013665.

Programs

  • Mathematica
    f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)
  • PARI
    J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));

Formula

a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)

A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

Original entry on oeis.org

1, 25, 217, 793, 3001, 5425, 16465, 25369, 52705, 75025, 159721, 172081, 369097, 411625, 651217, 811801, 1414945, 1317625, 2469241, 2379793, 3572905, 3993025, 6424177, 5505073, 9378001, 9227425, 12807289, 13056745, 20486761, 16280425, 28599361, 25977625, 34659457
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A084218, A350156.
Cf. A068970, A084220, A372964.
Cf. A001160, A008683.
Cf. A002117, A013664, A347328.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)
Dirichlet convolution of A334659 and A001160. - R. J. Mathar, Jul 14 2025

A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

Original entry on oeis.org

1, 13, 703, 61681, 9762501, 2140365529, 678222249307, 280379743338241, 150087010086914941, 99902428887422922553, 81402749386554449442711, 79477293980103609858493681, 91733330193268313783293023757, 123469159731637675342948027295569, 191751045863140709562160603031808243
Offset: 1

Views

Author

Leroy Quet, Jun 28 2005

Keywords

Examples

			sigma_4(4)/sigma_2(4) =
(1 + 2^4 + 4^4)/(1 + 2^2 + 4^2) = 13.

Crossrefs

Cf. A062755.
Cf. A057660, A084218, A084220, A372966.

Programs

  • Mathematica
    Table[DivisorSigma[2n, n^2]/DivisorSigma[n, n^2], {n, 10}] (* Ryan Propper, Apr 03 2007 *)
  • PARI
    a(n) = sigma(n^2, 2*n)/sigma(n^2, n); \\ Michel Marcus, Sep 06 2019

Formula

a(n) = Product_{p=primes} (Sum_{k=0..2*b(n, p)} p^(n*k)*(-1)^k), where p^b(n, p) is the highest power of p dividing n.
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, ... , x_n <= n} ( n/gcd(x_1, x_2, ... , x_n, n) )^n.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^n * sigma_{2*n}(d). (End)

Extensions

More terms from Ryan Propper, Apr 03 2007
More terms from Michel Marcus, Sep 06 2019

A371628 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 65, 757, 4225, 16001, 49205, 119365, 271489, 554797, 1040065, 1783541, 3198325, 4850977, 7758725, 12112757, 17392769, 24211265, 36061805, 47162485, 67604225, 90359305, 115930165, 148291397, 205517173, 250266001, 315313505, 404686153, 504317125, 595481825
Offset: 1

Views

Author

Seiichi Manyama, May 24 2024

Keywords

Links

Crossrefs

Cf. A373059, A371492.
Cf. A372963, A372964.
Cf. A084220.
Cf. A002117, A013662, A013665.

Programs

  • Mathematica
    f[p_, e_] := (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 24 2024 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 6)/sigma(d^2, 3));

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_6(d^2)/sigma_3(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)).
Dirichlet g.f.: zeta(s)*zeta(s-4)*zeta(s-6)/zeta(s-3)^2.
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(3)*zeta(7)/zeta(4)^2 = 1.034718122... . (End)
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