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 1-4 of 4 results.

A307793 a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 3, 7, 24, 49, 205, 411, 1668, 5011, 20095, 40191, 241372, 482745, 1931393, 7725627, 38629803, 77259607, 463562851, 927125703, 5562774334, 22251097753, 89004431205, 178008862411, 1424071142304, 4272213426961, 17088854190591, 68355416767375, 410132502535664, 820265005071329
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 29 2019

Keywords

Crossrefs

Cf. A000005, A007557, A060640, A307794, A319133.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[DivisorSigma[0, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[0, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 30}]
  • PARI
    a(n) = if (n==1, 1, sumdiv(n-1, d, numdiv(d)*a(d))); \\ Michel Marcus, Apr 29 2019

Formula

G.f.: x * (1 + Sum_{n>=1} tau(n)*a(n)*x^n/(1 - x^n)).
L.g.f.: -log(Product_{i>=1, j>=1} (1 - x^(i*j))^(a(i*j)/(i*j))) = Sum_{n>=1} a(n+1)*x^n/n.

A307967 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 6, 5, 11, 8, 14, 16, 20, 16, 37, 22, 34, 49, 44, 36, 90, 46, 73, 108, 80, 75, 181, 89, 121, 210, 151, 123, 334, 153, 197, 368, 227, 219, 567, 229, 313, 613, 365, 315, 871, 367, 461, 986, 519, 463, 1355, 534, 660, 1429, 756, 662, 1960, 794, 940, 2054
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2019

Keywords

Comments

Shifts 3 places left when inverse Moebius transform applied twice.

Crossrefs

Cf. A000005, A007557, A307982, A307993, A308083, A319133.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[DivisorSigma[0, (n - 3)/d] a[d], {d, Divisors[n - 3]}]; a[1] = a[2] = a[3] = 1; Table[a[n], {n, 1, 60}]

Formula

a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

A307982 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 6, 3, 11, 5, 15, 8, 19, 7, 36, 10, 31, 15, 60, 12, 56, 17, 97, 24, 72, 19, 170, 29, 94, 32, 229, 31, 156, 34, 334, 47, 182, 46, 471, 49, 218, 68, 658, 51, 314, 70, 797, 84, 354, 72, 1173, 93, 437, 98, 1353, 95, 576, 114, 1792, 131, 640, 116, 2243, 133
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2019

Keywords

Comments

Shifts 4 places left when inverse Moebius transform applied twice.

Crossrefs

Cf. A000005, A007557, A307967, A307994, A308083, A319133.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[DivisorSigma[0, (n - 4)/d] a[d], {d, Divisors[n - 4]}]; a[1] = a[2] = a[3] = a[4] = 1; Table[a[n], {n, 1, 65}]

Formula

a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

A308083 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 + x^5 * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 6, 3, 9, 5, 12, 11, 11, 11, 22, 14, 23, 19, 29, 24, 41, 25, 40, 41, 48, 43, 66, 45, 71, 67, 86, 68, 95, 73, 113, 110, 118, 107, 157, 115, 162, 148, 182, 159, 225, 164, 235, 229, 247, 227, 296, 244, 328, 297, 357, 298, 413, 352, 452, 409, 436, 415, 575
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2019

Keywords

Comments

Shifts 5 places left when inverse Moebius transform applied twice.

Crossrefs

Cf. A000005, A007557, A307967, A307982, A307995, A319133.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[DivisorSigma[0, (n - 5)/d] a[d], {d, Divisors[n - 5]}]; a[1] = a[2] = a[3] = a[4] = a[5] = 1; Table[a[n], {n, 1, 65}]

Formula

a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).
Showing 1-4 of 4 results.

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