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 1-5 of 5 results.

A353923 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} tau(n)*x^n, where tau = A000005.

Original entry on oeis.org

1, 2, 0, 3, -1, -1, -1, 9, 1, -7, 0, 4, -1, -19, 2, 94, -2, -57, 2, 81, -4, -186, 3, 226, 3, -632, -2, 1040, 1, -2060, -15, 10975, 17, -7720, -1, 13980, 9, -27595, -18, 50432, -10, -97582, 24, 191827, -17, -364695, 27, 580609, -37, -1338741, 45, 2658068, -11, -4909146, -98
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A000005, A320779, A328775, A353924, A353925, A353926.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, DivisorSigma[0, n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 55]

Formula

Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^n / (1 - x^n).

A353924 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} sigma(n)*x^n, where sigma = A000203.

Original entry on oeis.org

1, 3, 1, 6, -3, -6, -1, 40, 3, -50, -1, 73, 3, -315, 1, 1953, 1, -2117, 7, 5625, -13, -16116, 27, 33728, 29, -122648, 3, 351244, -25, -913057, -81, 5447169, 89, -7596153, 637, 22521844, -275, -61171056, -263, 177290760, -453, -487655426, -583, 1523295127, 8121, -4093188035
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A000203, A320780, A328776, A353923, A353925, A353926.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, DivisorSigma[1, n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 46]

Formula

Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^n / (1 - x^n)^2.

A353925 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} phi(n)*x^n, where phi = A000010.

Original entry on oeis.org

1, 1, 1, 1, 2, -2, 4, -3, 4, -6, 14, -18, 30, -32, 48, -73, 128, -182, 286, -394, 578, -804, 1386, -1936, 3172, -4298, 7102, -10202, 16414, -23798, 38056, -54997, 86658, -125180, 203396, -292650, 475536, -685314, 1100728, -1619952, 2601166, -3808952, 6114666, -9045400, 14352568
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A000010, A320778, A328774, A353923, A353924, A353926.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, EulerPhi[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 45]

A353949 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} mu(n)*x^n, where mu = A008683.

Original entry on oeis.org

1, -2, 0, -3, -1, 4, -4, -4, -7, 14, -10, -2, -5, 6, 19, -102, 70, -95, 138, -314, 174, 48, -70, -156, -797, 2028, -2048, 1989, -3479, 4277, -2080, -11462, 7923, -12448, 32218, -68038, 68683, -64844, 82847, -170573, -24942, 257846, -422887, 599115, -1225608, 2072993
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2022

Keywords

Crossrefs

Cf. A008683, A320781, A353926, A353927, A353945, A353947, A353948.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, MoebiusMu[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 46]

A353927 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} mu(n)*x^n, where mu = A008683.

Original entry on oeis.org

1, -1, 0, -1, -1, 2, -4, 4, -7, 8, -10, 9, -5, -6, 19, -40, 70, -110, 138, -158, 154, -93, -70, 355, -797, 1408, -2160, 2925, -3479, 3399, -2080, -1299, 7593, -17673, 32014, -49928, 68683, -82847, 82807, -53620, -24942, 176293, -422887, 777264, -1226688, 1710686, -2093347
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Comments

Inverse weigh transform of the Moebius function (A008683).

Crossrefs

Cf. A008683, A320781, A353926.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = MoebiusMu[n] - b[n, n - 1]; Table[a[n], {n, 1, 47}]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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