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 12 results. Next

A373040 a(n) = (A084218(n) - 1)/12.

Original entry on oeis.org

0, 1, 6, 17, 50, 79, 196, 273, 492, 651, 1210, 1247, 2366, 2549, 3656, 4369, 6936, 6397, 10830, 10267, 14314, 15731, 23276, 19935, 31300, 30759, 39858, 40197, 58870, 47529, 76880, 69905, 88336, 90169, 117846, 100877, 156066, 140791, 172724, 164123, 235340, 186083
Offset: 1

Views

Author

Hugo Pfoertner, May 20 2024

Keywords

Links

Crossrefs

Cf. A000290, A001157, A001159, A084218, A372966, A373039.
Cf. A002117, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 12; Array[a, 35] (* Amiram Eldar, Jan 03 2025 *)
  • PARI
    a(n) = (sigma(n^2, 4)/sigma(n^2, 2) - 1)/12

Formula

From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: (zeta(s-4)/zeta(s-2) - zeta(s))/12.
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(60*zeta(3)) = 0.0143771... . (End)

A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

Original entry on oeis.org

1, 57, 703, 3641, 15501, 40071, 117307, 233017, 512461, 883557, 1770231, 2559623, 4824613, 6686499, 10897203, 14913081, 24132657, 29210277, 47039023, 56439141, 82466821, 100903167, 148023723, 163810951, 242203001, 275002941, 373584043
Offset: 1

Views

Author

Benoit Cloitre, Jun 21 2003

Keywords

Links

Crossrefs

Cf. A001158, A013665, A013954.
Cf. A057660, A084218.

Programs

  • Maple
    with(numtheory): a:=n->sigma[6](n^2)/sigma[3](n^2): seq(a(n),n=1..30); # Muniru A Asiru, Oct 09 2018
  • Mathematica
    Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* Harvey P. Dale, May 02 2012 *)
f[p_, e_] := (p^(6*e + 3) + 1)/(p^3 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 13 2020 *)
  • PARI
    a(n)=sumdiv(n^2,d,d^6)/sumdiv(n^2,d,d^3)
  • PARI
    a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ Michel Marcus, Oct 09 2018

Formula

Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = 90*zeta(7)/(7*Pi^4) = 0.133093... . - Amiram Eldar, Oct 30 2022
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^3.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_6(d). (End)

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 29, 235, 925, 3101, 6815, 16759, 29597, 57097, 89929, 160931, 217375, 371125, 486011, 728735, 947101, 1419569, 1655813, 2475739, 2868425, 3938365, 4666999, 6435815, 6955295, 9690601, 10762625, 13874563, 15502075, 20510309, 21133315, 28628191, 30307229, 37818785
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Cf. A068963, A084218, A372963.
Cf. A001160, A008683.
Cf. A013662, A013664, A088246.
Cf. A350156, A372964.
Cf. A371492, A372952.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 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)^2*sigma(d, 5));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 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)

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

A373007 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) )^2.

Original entry on oeis.org

1, 125, 2179, 15997, 78101, 272375, 823495, 2047613, 4765465, 9762625, 19487051, 34857463, 62748349, 102936875, 170182079, 262094461, 410338385, 595683125, 893871379, 1249381697, 1794395605, 2435881375, 3404824919, 4461748727, 6101640601, 7843543625, 10422071947
Offset: 1

Views

Author

Seiichi Manyama, May 25 2024

Keywords

Links

Crossrefs

Cf. A371491, A371878, A373103, A373105.
Cf. A068963, A084218, A372962, A372963.
Cf. A013664, A013666.

Programs

  • Mathematica
    f[p_, e_] := (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 27] (* Amiram Eldar, May 25 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 7));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_7(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(6) = Pi^2/10 = 0.986960440108... . (End)

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)

A371492 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 17, 91, 289, 701, 1547, 2647, 4769, 7705, 11917, 15731, 26299, 30421, 44999, 63791, 77473, 87857, 130985, 136459, 202589, 240877, 267427, 290951, 433979, 448201, 517157, 633187, 764983, 729989, 1084447, 951391, 1248929, 1431521, 1493569, 1855547, 2226745
Offset: 1

Views

Author

Seiichi Manyama, May 24 2024

Keywords

Links

Crossrefs

Cf. A084218, A373060.
Cf. A372952, A372962.
Cf. A373059, A371628.
Cf. A002117, A013661, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 36] (* Amiram Eldar, May 24 2024 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^2*sigma(d^2, 4)/sigma(d^2, 2));

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_4(d^2)/sigma_2(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)).
Dirichlet g.f.: zeta(s)*zeta(s-3)*zeta(s-4)/zeta(s-2)^2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)*zeta(5)/zeta(3)^2 = 1.180448217... . (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

A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).

Original entry on oeis.org

1, 22, 105, 394, 745, 2310, 2737, 6490, 8817, 16390, 15961, 41370, 30745, 60214, 78225, 104602, 88417, 193974, 137161, 293530, 287385, 351142, 291985, 681450, 469345, 676390, 717081, 1078378, 731641, 1720950, 953281, 1676698, 1675905, 1945174, 2039065, 3473898
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A007434, A084218.
Cf. A062952, A373133, A373135.
Cf. A002117, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1); 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=2, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

Formula

a(n) = Sum_{d|n} J_2(d) * sigma(d^2), where the Jordan totient function J_2(n) = A007434(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3) * zeta(5) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^4 - 1/p^5 + 1/p^6) = 1.489916841715667671605... . (End)
Showing 1-10 of 12 results. Next

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