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

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

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)

A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965
Offset: 1

Views

Author

Seiichi Manyama, May 21 2024

Keywords

Links

Crossrefs

Row sums of A350900.
Cf. A057660, A350156.
Cf. A001620, A002117, A013661, A073002, A244115, A306016.

Programs

  • Mathematica
    f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ Amiram Eldar, May 27 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (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)

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

A373129 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( n/gcd(x_1, x_2, n) ).

Original entry on oeis.org

1, 10, 33, 94, 145, 330, 385, 814, 969, 1450, 1441, 3102, 2353, 3850, 4785, 6766, 5185, 9690, 7201, 13630, 12705, 14410, 12673, 26862, 18745, 23530, 26889, 36190, 25201, 47850, 30721, 55150, 47553, 51850, 55825, 91086, 51985, 72010, 77649, 118030, 70561, 127050
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A007434, A013662, A064987, A350156.

Programs

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

Formula

a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2 ).
a(n) = Sum_{d|n} J_2(d) * sigma(d), where the Jordan totient function J_2(n) = A007434(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e+2)*(p+1) - p^(2*e)*(p^2+p+1) + p)/(p^3-1).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.41347980562116709395... . (End)
Showing 1-7 of 7 results.

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