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.

Previous Showing 11-17 of 17 results.

A290137 Number of compositions (ordered partitions) of n into nonprime squarefree parts (A000469).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 22, 30, 41, 55, 73, 96, 128, 173, 235, 317, 426, 570, 763, 1023, 1375, 1848, 2484, 3337, 4482, 6017, 8077, 10843, 14562, 19560, 26276, 35292, 47392, 63632, 85443, 114741, 154098, 206957, 277941, 373254, 501244, 673121, 903945, 1213935, 1630246, 2189330
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 20 2017

Keywords

Examples

			a(8) = 4 because we have [6, 1, 1], [1, 6, 1], [1, 1, 6] and [1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000469, A239508, A239509, A280194.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A000469(k)).

A301500 Number of compositions (ordered partitions) of n into squarefree parts (A005117) such that no two adjacent parts are equal (Carlitz compositions).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 11, 15, 25, 45, 69, 115, 193, 309, 513, 849, 1387, 2291, 3771, 6189, 10195, 16773, 27579, 45391, 74675, 122837, 202111, 332507, 547011, 899949, 1480583, 2435803, 4007361, 6592863, 10846405, 17844319, 29357197, 48297813, 79458705, 130724101, 215064673
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2018

Keywords

Examples

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

Links

Crossrefs

Cf. A003242, A005117, A008683, A280194, A301428, A301501.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k/(1 + x^k)), where mu() is the Moebius function (A008683).

A331982 Number of compositions (ordered partitions) of n into distinct odd squarefree parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 2, 1, 4, 6, 2, 7, 4, 7, 4, 13, 30, 13, 8, 25, 32, 31, 56, 37, 82, 42, 104, 168, 128, 175, 152, 181, 226, 307, 252, 439, 326, 691, 372, 943, 1190, 1069, 1238, 1435, 2056, 1806, 2102, 2185, 3664, 2550, 4480, 3175, 6090, 3781, 7628, 9691
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2020

Keywords

Examples

			a(8) = 4 because we have [7, 1], [5, 3], [3, 5] and [1, 7].

Links

Crossrefs

Cf. A056911, A087188, A134337, A134345, A280194, A280198, A331846, A331981.

A281812 Expansion of Sum_{i>=1} mu(i)^2*x^i / (1 - Sum_{j>=1} mu(j)^2*x^j)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 44, 99, 218, 473, 1012, 2144, 4504, 9395, 19482, 40189, 82534, 168829, 344145, 699334, 1417146, 2864510, 5776889, 11626101, 23353272, 46827677, 93747221, 187399328, 374092162, 745817021, 1485138398, 2954041789, 5869650947, 11651500427, 23107388495, 45787040997, 90652188078, 179340159228
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2017

Keywords

Comments

Total number of parts in all compositions (ordered partitions) of n into squarefree parts (A005117).

Examples

			a(4) = 19 because we have [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

Links

Crossrefs

Cf. A005117, A008683, A121304, A280194.

Programs

  • Mathematica
    nmax = 36; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i, {i, 1, nmax}]/(1 - Sum[MoebiusMu[j]^2 x^j, {j, 1, nmax}])^2, {x, 0, nmax}], x]]

Formula

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

A329099 Expansion of 1 / (1 + Sum_{k>=1} mu(k)^2 * x^k).

Original entry on oeis.org

1, -1, 0, 0, 1, -2, 1, 0, 2, -4, 2, 0, 4, -10, 7, 0, 7, -23, 22, -6, 14, -51, 59, -24, 31, -113, 152, -80, 66, -244, 383, -253, 166, -521, 930, -746, 460, -1133, 2219, -2082, 1314, -2494, 5208, -5607, 3788, -5622, 12037, -14608, 10830, -13145, 27618, -37089, 30350, -31914, 63248, -92290
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

Crossrefs

Cf. A002121, A005117, A008683, A280194.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 + Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[SquareFreeQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 55}]

Formula

G.f.: 1 / (1 + Sum_{k>=1} x^A005117(k)).

A347783 Number of compositions (ordered partitions) of n into at most 6 squarefree parts.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 27, 51, 92, 155, 247, 376, 558, 805, 1118, 1495, 1962, 2551, 3282, 4127, 5106, 6286, 7750, 9442, 11326, 13459, 16037, 19011, 22277, 25804, 29939, 34692, 39938, 45455, 51736, 58848, 66696, 74836, 83927, 94114, 105372, 116992, 129902, 144223
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A341066, A347777, A347778, A347779, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Select[Range@n,SquareFreeQ]],1],{n,0,43}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A369220 Number of compositions (ordered partitions) of n into squarefree parts not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 8, 13, 21, 34, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 4586976, 8662591, 16359466, 30895160, 58346092, 110187694, 208091537, 392984789, 742159180, 1401581598, 2646913261, 7359931330, 14066178853
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 16 2024

Keywords

Crossrefs

Cf. A280194, A364526.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[Boole[SquareFreeQ[k]] x^k, {k, 1, Floor[Sqrt[n]]}]), {x, 0, n}], {n, 0, 37}]
Previous Showing 11-17 of 17 results.

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