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.

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A284833 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j=1..i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 3, 7, 6, 11, 8, 17, 12, 22, 21, 28, 27, 41, 35, 53, 52, 66, 66, 90, 85, 112, 114, 140, 143, 182, 180, 219, 236, 269, 291, 342, 353, 417, 444, 508, 540, 625, 657, 751, 812, 901, 974, 1097, 1168, 1313, 1414, 1562, 1684, 1874, 2008, 2219, 2397, 2626, 2832, 3121, 3341, 3668, 3956, 4305, 4650
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into prime parts.

Examples

			a(10) = 11 because we have [7, 3], [5, 5], [5, 3, 2], [3, 3, 2, 2], [2, 2, 2, 2, 2] and 1 + 2 + 1 + 2 + 5 = 11.

Links

Crossrefs

Cf. A000607, A046746, A084993, A092311, A281544, A284827.

Programs

  • Mathematica
    nmax = 65; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^66); concat([0], Vec(sum(i=1, 66, x^prime(i)/(1 - x^prime(i)) * prod(j=1,i, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j=1..i} 1/(1 - x^prime(j)).

A284834 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j=2..i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 2, 5, 4, 4, 9, 5, 6, 12, 8, 11, 17, 12, 14, 23, 19, 21, 29, 27, 29, 41, 37, 36, 56, 49, 55, 72, 62, 74, 91, 90, 96, 116, 117, 125, 155, 149, 162, 195, 194, 215, 246, 248, 270, 311, 324, 344, 389, 406, 435, 494, 509, 546, 615, 636, 694, 763, 787, 861, 942, 994, 1063
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into odd prime parts (A065091).

Examples

			a(16) = 5 because we have [13, 3], [11, 5], [7, 3, 3, 3], [5, 5, 3, 3] and 1 + 1 + 1 + 2 = 5.

Links

Crossrefs

Cf. A046746, A065091, A084993, A092311, A099773, A284828.

Programs

  • Mathematica
    nmax = 64; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, 2, i}], {i, 2, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^70); concat([0, 0], Vec(sum(i=2, 70, x^prime(i)/(1 - x^prime(i)) * prod(j=2,i, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j=2..i} 1/(1 - x^prime(j)).

A284835 Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j=1..i} 1/(1 - mu(j)^2*x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 5, 8, 11, 18, 22, 31, 39, 53, 64, 87, 104, 134, 165, 205, 248, 310, 368, 455, 545, 659, 784, 947, 1117, 1337, 1579, 1872, 2197, 2604, 3036, 3570, 4168, 4866, 5661, 6599, 7633, 8859, 10236, 11831, 13625, 15715, 18036, 20728, 23761, 27211, 31106, 35560, 40533, 46221, 52596, 59813, 67912, 77090, 87343
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into squarefree parts (A005117)

Examples

			a(5) = 11 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 1 + 1 + 2 + 1 + 5 = 11.

Links

Crossrefs

Cf. A005117, A008683, A046746, A073576, A092311, A281572, A284829.

Programs

  • Mathematica
    nmax = 55; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i) Product[1/(1 - MoebiusMu[j]^2 x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^56); Vec(sum(i=1, 56, moebius(i)^2*x^i/(1 - x^i) * prod(j=1, i, 1/(1 - moebius(j)^2*x^j)))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j=1..i} 1/(1 - mu(j)^2*x^j).

A284836 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j=1..i} 1/(1 - x^(j^2)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 19, 21, 22, 23, 29, 31, 34, 35, 42, 44, 47, 48, 56, 60, 63, 67, 76, 80, 83, 87, 99, 103, 108, 112, 130, 134, 139, 143, 162, 169, 174, 180, 200, 213, 218, 224, 248, 262, 272, 278, 306, 320, 337, 343, 372, 390, 408, 419, 449, 471, 489, 508, 544, 567, 591, 611, 654, 677, 705
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into squares (A000290).

Examples

			a(9) = 13 because we have [9], [4, 4, 1], [4, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 2 + 1 + 9 = 13.

Links

Crossrefs

Cf. A000290, A001156, A046746, A092311, A281541, A284830.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2) Product[1/(1 - x^j^2), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^71); Vec(sum(i=1, 71, x^(i^2)/(1 - x^(i^2)) * prod(j=1, i, 1/(1 - x^(j^2))))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j=1..i} 1/(1 - x^(j^2)).

A284837 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 32, 34, 35, 36, 37, 38, 43, 44, 45, 47, 48, 49, 50, 51, 57, 58, 59, 61, 62, 63, 64, 65, 72, 73, 74, 76, 77, 78, 81, 82, 90, 91, 92, 94, 95, 96, 99, 100, 110, 111, 112, 114, 115, 116, 119, 120, 131, 132, 133, 135
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into cubes (A000578).

Examples

			a(10) = 11 because we have [8, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 10 = 11.

Links

Crossrefs

Cf. A000578, A003108, A046746, A092311, A281613, A284831.

Programs

  • Mathematica
    nmax = 75; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3) Product[1/(1 - x^j^3), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).
Previous Showing 11-15 of 15 results.

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