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 1-3 of 3 results.

A366450 a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

Original entry on oeis.org

1, -2, -1, -4, 1, 2, -2, -8, -3, -2, 1, 4, 4, 4, -1, -16, -2, 6, 0, -4, 2, -2, -1, 8, 5, -8, -9, 8, 0, 2, 7, -32, -1, 4, -2, 12, 3, 0, -4, -8, -8, -4, -6, -4, -3, 2, 8, 16, -14, -10, 2, -16, -6, 18, 1, 16, 0, 0, 5, 4, 12, -14, 6, -64, 4, 2, -7, 8, 1, 4, -3, 24, 4, -6, -5, 0, -2, 8, -10, -16, -27, 16, -6, -8
Offset: 1

Views

Author

Mats Granvik, Oct 10 2023

Keywords

Comments

It appears that: a(A005117(n)) = A006571(A005117(n)), verified up to n = 98. But also a(76) = A006571(76), a(116) = A006571(116) and a(152) = A006571(152). 76 = 19*2^2, 116 = 29*2^2 and 152 = 19*2^3.

Crossrefs

Cf. A366362, A023900, A006571, A005117, A002070, A366362.

Programs

  • Mathematica
    nn = 84; f = x^3 - x^2 - y^2 - y; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Monitor[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0]*g[k]/n, {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}], n]
  • PARI
    a(n) = sum(k=1, n, my(z=sumdivmult(k, d, d*moebius(d))); sum(y=1, n, sum(x=1, n, if (gcd(x^3 - x^2 - y^2 - y, n)==k, z/n)))); \\ Michel Marcus, Oct 10 2023

Formula

a(n) = Sum_{k=1..n} A366362(n,k)*A023900(k)/n.

A366346 a(n) = A002070(n) + A036689(n).

Original entry on oeis.org

0, 5, 21, 40, 111, 160, 270, 342, 505, 812, 937, 1335, 1632, 1800, 2170, 2750, 3427, 3672, 4415, 4967, 5260, 6152, 6800, 7847, 9305, 10102, 10490, 11360, 11782, 12665, 16010, 17012, 18625, 19192, 22042, 22652, 24485, 26410, 27710, 29750, 31847, 32587, 36307
Offset: 1

Views

Author

Mats Granvik, Oct 07 2023

Keywords

Crossrefs

Cf. A366362, A002070, A036689, A001248, A272196.

Programs

  • Mathematica
    b[n_] := If[n < 1, 0, With[{m = Prime@n}, SeriesCoefficient[q (Product[(1 - q^(11 k)), {k, Ceiling[m/11]}] Product[1 - q^k, {k, m}])^2, {q, 0, m}]]]; Table[Prime[n] (Prime[n] - 1) + b[n], {n, 1, 43}] (* after Michael Somos in A002070, Jul 04 2011 *)

Formula

a(n) = A002070(n) + A036689(n).
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} [GCD(f(x,y), prime(n)) = 1],
a(n) = Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} (1 - [MOD(f(x,y), prime(n)) = 0]) where f(x,y) = x^3 - x^2 - y^2 - y, in the last two formulas.
a(n) = A001248(n) - A272196(n), for n > 1.
a(n) = A366362(prime(n), 1).

A366417 a(n) = A006571(A005117(n)).

Original entry on oeis.org

1, -2, -1, 1, 2, -2, -2, 1, 4, 4, -1, -2, 0, 2, -2, -1, -8, 0, 2, 7, -1, 4, -2, 3, 0, -4, -8, -4, -6, 2, 8, 2, -6, 1, 0, 0, 5, 12, -14, 4, 2, -7, 1, 4, -3, 4, -6, -2, 8, -10, 16, -6, -2, 12, 0, 15, -8, -7, -16, 0, -7, 2, -4, -16, 2, 12, 18, 10, -2, -3, 9, 0, -1
Offset: 1

Views

Author

Mats Granvik, Oct 10 2023

Keywords

Crossrefs

Cf. A006571, A366450, A005117, A002070, A023900, A191898, A366362.

Programs

  • Mathematica
    nn = 73; squareFree = Select[Range[8*nn], SquareFreeQ]; b[n_] := SeriesCoefficient[q (Product[(1 - q^k), {k, 11, n, 11}] Product[1 - q^k, {k, n}])^2, {q, 0, n}]; Table[b[squareFree[[n]]], {n, 1, nn}]

Formula

a(n) = A006571(A005117(n)).
Conjecture: a(n) = A366450(A005117(n)), verified up to n = 98.
Showing 1-3 of 3 results.

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