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 1-9 of 9 results.

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

A350156 Inverse Moebius transform of A000056.

Original entry on oeis.org

1, 7, 25, 55, 121, 175, 337, 439, 673, 847, 1321, 1375, 2185, 2359, 3025, 3511, 4897, 4711, 6841, 6655, 8425, 9247, 12145, 10975, 15121, 15295, 18169, 18535, 24361, 21175, 29761, 28087, 33025, 34279, 40777, 37015, 50617, 47887, 54625, 53119, 68881, 58975, 79465, 72655, 81433, 85015
Offset: 1

Views

Author

Werner Schulte, Jan 19 2022

Keywords

Comments

Let f be an arbitrary arithmetic function. Define the sequence a(f; n) by a(f; n) = Sum_{i=1..n, k=1..n} f(n / gcd(gcd(i,k),n)) for n > 0. Then a(f; n) equals inverse Moebius transform of f(n) * A007434(n) for n > 0; if f is multiplicative then a(f; n) is multiplicative; this sequence uses f(n) = n (see formula section).

Crossrefs

Column k=2 of A372968.
Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.
Cf. A372962, A372964.

Programs

  • Mathematica
    f[p_, e_] := p^(3*e) - (p - 1)*(p^(3*e) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Jan 19 2022 *)
  • Python
    from math import prod
from sympy import factorint
def A350156(n): return prod((q:=p**(3*e))-(p-1)*(q-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Formula

Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).
a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.
Dirichlet convolution with A000010 equals A000578.
Dirichlet convolution of A001158 and A055615.
Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - Amiram Eldar, Oct 16 2022
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2. - Seiichi Manyama, May 25 2024

A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

Original entry on oeis.org

1, 15, 79, 239, 621, 1185, 2395, 3823, 6397, 9315, 14631, 18881, 28549, 35925, 49059, 61167, 83505, 95955, 130303, 148419, 189205, 219465, 279819, 302017, 388121, 428235, 518155, 572405, 707253, 735885, 923491, 978671, 1155849, 1252575, 1487295, 1528883
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Links

Crossrefs

Column k=3 of A372968.
Cf. A001159, A008683.
Cf. A013662, A013663.
Cf. A371492, A372962.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1); 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*sigma(d,4));

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_4(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1).
Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(5)/zeta(4) = 0.958057374... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d) * 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, x_2, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

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

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)

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)

A373103 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) )^4.

Original entry on oeis.org

1, 497, 19603, 254449, 1952501, 9742691, 40351207, 130277873, 385845769, 970392997, 2357933051, 4987963747, 10604470813, 20054549879, 38274877103, 66702270961, 118587792977, 191765347193, 322687567459, 496811926949, 791004710821, 1171892726347, 1801152381623
Offset: 1

Views

Author

Seiichi Manyama, May 25 2024

Keywords

Links

Crossrefs

Cf. A371491, A371878, A373007, A373105.
Cf. A350156, A372962, A372964.
Cf. A013664, A013668.

Programs

  • Mathematica
    f[p_, e_] :=  (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 25 2024 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 9));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^8*sigma(d^2, 8)/sigma(d^2, 4));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_9(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^8 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1).
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = zeta(10)/zeta(6) = Pi^4/99 = 0.983930212464... . (End)

A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).

Original entry on oeis.org

1, 50, 339, 1786, 3845, 16950, 19495, 58682, 85281, 192250, 176891, 605454, 401869, 974750, 1303455, 1890106, 1507985, 4264050, 2612899, 6867170, 6608805, 8844550, 6727799, 19893198, 12109345, 20093450, 20802003, 34818070, 21241949, 65172750, 29581471, 60581690
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A059376, A372962.
Cf. A373130, A373133.
Cf. A013664.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1); 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=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d^2), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4) = 1.67666099579383196077... . (End)
Showing 1-9 of 9 results.

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