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.

A339218 Number of partitions of n into prime parts where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 3, 1, 4, 3, 5, 1, 6, 4, 8, 6, 9, 5, 12, 9, 14, 11, 17, 13, 22, 17, 24, 21, 31, 26, 37, 31, 42, 39, 52, 46, 61, 56, 71, 67, 84, 79, 100, 95, 114, 111, 135, 131, 158, 154, 180, 180, 212, 209, 244, 244, 280, 283, 324, 325, 372, 378, 426, 434, 487
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2020

Keywords

Examples

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

Links

Crossrefs

Cf. A000040, A000607, A007690, A161077.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1 + x^(2 Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(2*prime(k)) / (1 - x^prime(k))).

A161078 Number of partitions of n into primes or 1 where every part appears at least 3 times.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 10, 8, 9, 14, 13, 16, 21, 19, 23, 30, 31, 34, 42, 44, 49, 60, 61, 68, 83, 85, 96, 111, 115, 129, 149, 158, 173, 196, 210, 229, 260, 275, 301, 339, 359, 392, 436, 462, 505, 559, 594, 645, 708, 755, 817, 895, 952, 1026, 1123, 1194, 1287
Offset: 1

Views

Author

R. H. Hardin, Jun 02 2009

Keywords

Examples

			a(12)=6 because we have 3333, 333111, 2^6, 22221111, 2221^6, and 1^(12). - _Emeric Deutsch_, Jun 27 2009

Links

Crossrefs

Cf. A161077.

Programs

  • Maple
    g := -1+(1+x^3/(1-x))*(product(1+x^(3*ithprime(j))/(1-x^ithprime(j)), j = 1 .. 20)): gser := series(g, x = 0, 75): seq(coeff(gser, x, n), n = 2 .. 65); # Emeric Deutsch, Jun 27 2009

Formula

G.f.: -1+(1+x^3/(1-x))*Product_{j>=1} ( 1+x^(3*p(j))/(1-x^(p(j))) ), where p(j) is the j-th prime. - Emeric Deutsch, Jun 27 2009

Extensions

Definition edited to "primes or 1" by R. H. Hardin, Jun 22 2009

A339241 Number of partitions of n into prime power parts (including 1) where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 15, 11, 21, 19, 31, 27, 46, 40, 63, 60, 88, 83, 124, 117, 166, 165, 224, 222, 303, 301, 399, 407, 525, 537, 691, 707, 893, 929, 1153, 1202, 1485, 1550, 1890, 1992, 2400, 2534, 3040, 3212, 3818, 4059, 4781, 5089, 5972, 6359, 7412
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2020

Keywords

Examples

			a(6) = 4 because we have [3, 3], [2, 2, 2], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000961, A007690, A023893, A161077, A339218, A339242.

Programs

  • Mathematica
    nmax = 56; CoefficientList[Series[(1 + x^2/(1 - x)) Product[1 + Boole[PrimePowerQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1 + x^2 / (1 - x)) * Product_{p prime, k>=1} (1 + x^(2*p^k) / (1 - x^(p^k))).

A339242 Number of partitions of n into prime power parts (1 excluded) where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 5, 1, 6, 3, 9, 2, 12, 4, 15, 8, 21, 8, 28, 13, 34, 20, 45, 23, 59, 34, 73, 47, 92, 57, 119, 78, 145, 103, 182, 128, 229, 166, 277, 213, 344, 265, 427, 334, 513, 420, 629, 517, 771, 641, 923, 794, 1120, 967, 1355, 1182, 1618, 1447, 1946, 1745
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2020

Keywords

Examples

			a(12) = 5 because we have [4, 4, 4], [4, 4, 2, 2], [3, 3, 3, 3], [3, 3, 2, 2, 2] and [2, 2, 2, 2, 2, 2].

Links

Crossrefs

Cf. A007690, A023894, A161077, A246655, A339218, A339241.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Product[1 + Boole[PrimePowerQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{p prime, k>=1} (1 + x^(2*p^k) / (1 - x^(p^k))).

A339219 Number of partitions of n into nonprime parts where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 2, 4, 4, 6, 4, 8, 6, 11, 8, 11, 11, 17, 11, 19, 18, 25, 20, 32, 26, 42, 32, 46, 43, 63, 47, 72, 66, 90, 74, 110, 94, 137, 115, 155, 145, 203, 161, 235, 212, 283, 244, 339, 298, 413, 356, 472, 437, 589, 496, 681, 625, 810, 718, 962
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2020

Keywords

Examples

			a(12) = 4 because we have [6, 6], [4, 4, 4], [4, 4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A002095, A007690, A018252, A161077, A339218.

Programs

Formula

G.f.: Product_{k>=1} (1 + x^(2*A018252(k)) / (1 - x^A018252(k))).
Showing 1-5 of 5 results.

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