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 1-7 of 7 results.

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

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

A372968 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} n/gcd(x_1, x_2, ..., x_k, n).

Original entry on oeis.org

1, 1, 3, 1, 7, 7, 1, 15, 25, 11, 1, 31, 79, 55, 21, 1, 63, 241, 239, 121, 21, 1, 127, 727, 991, 621, 175, 43, 1, 255, 2185, 4031, 3121, 1185, 337, 43, 1, 511, 6559, 16255, 15621, 7471, 2395, 439, 61, 1, 1023, 19681, 65279, 78121, 45801, 16801, 3823, 673, 63
Offset: 1

Views

Author

Seiichi Manyama, May 18 2024

Keywords

Examples

			Square array begins:
   1,   1,    1,     1,      1,      1, ...
   3,   7,   15,    31,     63,    127, ...
   7,  25,   79,   241,    727,   2185, ...
  11,  55,  239,   991,   4031,  16255, ...
  21, 121,  621,  3121,  15621,  78121, ...
  21, 175, 1185,  7471,  45801, 277495, ...

Crossrefs

Columns k=1..5 give A057660, A350156, A372952, A372961, A371878.
Main diagonal gives A372969.

Programs

  • Mathematica
    f[p_, e_, k_] := (p^((k + 1)*e + k + 1) - p^((k + 1)*e + 1) + p - 1)/(p^(k + 1) - 1); T[1, k_] := 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 25 2024 *)
  • PARI
    T(n, k) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, k+1));

Formula

T(n,k) = Sum_{d|n} mu(n/d) * (n/d) * sigma_{k+1}(d).
T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} ( gcd(x_1, x_2, ..., x_{k-1}, n)/gcd(x_1, x_2, ..., x_k, n) )^k.
From Amiram Eldar, May 25 2024: (Start)
T(n,k) for a given k is multiplicative with T(p^e, k) = (p^((k+1)*(e+1)) - p^((k+1)*e+1) + p - 1)/(p^(k+1)-1).
Dirichlet g.f. of T(n, k) for a given k: zeta(s)*zeta(s-k-1)/zeta(s-1).
Sum_{m=1..n} T(m, k) ~ c * n^(k+2) / (k+2), where c = zeta(k+2)/zeta(k+1). (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)

A373060 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, n)/gcd(x_1, x_2, x_3, n).

Original entry on oeis.org

1, 11, 43, 115, 221, 473, 631, 1139, 1609, 2431, 2531, 4945, 4213, 6941, 9503, 10867, 9521, 17699, 13339, 25415, 27133, 27841, 23783, 48977, 39721, 46343, 55555, 72565, 47909, 104533, 58591, 100979, 108833, 104731, 139451, 185035, 99901, 146729, 181159, 251719
Offset: 1

Views

Author

Seiichi Manyama, May 21 2024

Keywords

Links

Crossrefs

Cf. A372952, A373061.

Programs

  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd(i, n)/gcd([i, j, k, n]))));

A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).

Original entry on oeis.org

1, 22, 105, 414, 745, 2310, 2737, 7134, 9231, 16390, 15961, 43470, 30745, 60214, 78225, 118238, 88417, 203082, 137161, 308430, 287385, 351142, 291985, 749070, 481245, 676390, 767391, 1133118, 731641, 1720950, 953281, 1924574, 1675905, 1945174, 2039065, 3821634
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A059376, A328259, A372952.
Cf. A373131, A373133.
Cf. A013661, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + 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=3, m=1) = 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, x_2, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d), where the Jordan totient function J_3(n) = A059376(n)
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.54488120152452251241... . (End)

A373061 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).

Original entry on oeis.org

1, 9, 31, 77, 141, 279, 379, 637, 877, 1269, 1431, 2387, 2341, 3411, 4371, 5181, 5169, 7893, 7183, 10857, 11749, 12879, 12651, 19747, 18041, 21069, 24043, 29183, 25173, 39339, 30691, 41789, 44361, 46521, 53439, 67529, 51949, 64647, 72571, 89817, 70521, 105741
Offset: 1

Views

Author

Seiichi Manyama, May 21 2024

Keywords

Links

Crossrefs

Cf. A372952, A373060.
Cf. A002117, A013661, A013662.

Programs

  • Mathematica
    f[p_, e_] := (p^(3*e)*(p+1)^3 - p^(2*e)*(p^2+p+1) + 1)/((p^2+p+1)*(p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, n])/gcd([i, j, k, n]))));

Formula

From Amiram Eldar, May 31 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e)*(p+1)^3 - p^(2*e)*(p^2+p+1) + 1)/((p^2+p+1)*(p+1)).
Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(s-3)/zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(4) / zeta(3)^2 = Pi^6/(540*zeta(3)^2) = 1.232126852811... . (End)
Showing 1-7 of 7 results.

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