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.

A331926 Number of compositions (ordered partitions) of n into distinct prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 8, 3, 10, 8, 14, 31, 10, 33, 16, 38, 40, 61, 138, 69, 48, 98, 190, 121, 308, 128, 340, 270, 472, 991, 572, 885, 534, 446, 888, 1872, 914, 1927, 1084, 2300, 2058, 4303, 6508, 3759, 2246, 4856, 8238, 6889, 12630, 6368, 8708
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Links

Crossrefs

Cf. A008578, A023360, A034891, A036497, A219107, A280917.

Programs

  • PARI
    a(n)={subst(serlaplace(y^0*polcoef(prod(k=1, n, 1 + if(k==1 || isprime(k), y*x^k) + O(x*x^n)), n)), y, 1)} \\ Andrew Howroyd, Feb 01 2020

A347703 Number of compositions (ordered partitions) of n into at most 4 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 20, 28, 36, 43, 53, 60, 68, 70, 79, 84, 103, 105, 124, 118, 147, 144, 183, 165, 207, 180, 243, 213, 280, 215, 290, 222, 334, 258, 377, 260, 411, 299, 471, 324, 491, 302, 530, 346, 595, 377, 639, 380, 705, 435, 766, 463, 819, 456, 886, 506, 942
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A008578, A280917, A341983, A347741, A347744, A347745, A347760, A347761.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Join[{1},Prime@Range@PrimePi@n]],1],{n,0,56}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A347760 Number of compositions (ordered partitions) of n into at most 5 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 43, 66, 93, 124, 160, 198, 240, 274, 315, 353, 415, 464, 538, 577, 669, 718, 850, 887, 1031, 1043, 1238, 1250, 1495, 1435, 1692, 1584, 1943, 1817, 2251, 2011, 2529, 2261, 2939, 2561, 3287, 2720, 3596, 3005, 4077, 3304, 4505, 3545, 4995, 3966
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A008578, A280917, A341984, A347703, A347742, A347744, A347745, A347761.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Join[{1},Prime@Range@PrimePi@n]],1],{n,0,50}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A347761 Number of compositions (ordered partitions) of n into at most 6 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 49, 87, 143, 220, 316, 435, 576, 739, 911, 1100, 1297, 1541, 1798, 2113, 2405, 2785, 3136, 3648, 4061, 4670, 5080, 5828, 6301, 7261, 7692, 8751, 9059, 10379, 10681, 12329, 12435, 14354, 14335, 16847, 16673, 19588, 18866, 22247, 21257
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2021

Keywords

Crossrefs

Cf. A008578, A280917, A341985, A347703, A347743, A347744, A347745, A347760.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Join[{1},Prime@Range@PrimePi@n]],1],{n,0,45}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

A281422 Expansion of 1/(1 - Sum_{k>=1} x^prime(prime(k))).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 4, 1, 3, 6, 2, 8, 9, 5, 16, 13, 14, 30, 20, 33, 51, 37, 72, 84, 76, 142, 141, 164, 264, 247, 344, 473, 462, 694, 836, 903, 1344, 1494, 1799, 2520, 2734, 3566, 4638, 5145, 6951, 8489, 9875, 13295, 15632, 19110, 25037, 29130, 36919, 46732, 54969, 70798, 87026, 104653, 134585, 162550
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into primes with prime subscripts (A006450).

Examples

			a(11) = 4 because we have [11], [5, 3, 3], [3, 5, 3] and [3, 3, 5], where 3 = prime(2) = prime(prime(1)), 5 = prime(3) = prime(prime(2)) and 11 = prime(5) = prime(prime(3)).

Links

Crossrefs

Cf. A006450, A023360, A271368, A278912, A280917.

Programs

  • Mathematica
    nmax = 64; CoefficientList[Series[1/(1 - Sum[x^Prime[Prime[k]], {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

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

A282971 Number of compositions (ordered partitions) of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 9, 11, 15, 18, 24, 29, 37, 48, 58, 78, 92, 124, 149, 195, 243, 308, 393, 490, 629, 786, 1004, 1263, 1603, 2024, 2564, 3239, 4106, 5184, 6571, 8301, 10508, 13298, 16807, 21296, 26895, 34082, 43060, 54528, 68952, 87245, 110392, 139622, 176696, 223484, 282798, 357731
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 25 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into primes congruent to 1 or 2 mod 4.
Conjecture: every number > 16 is the sum of at most 4 primes of form x^2 + y^2.

Examples

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

Links

Crossrefs

Cf. A002124, A002313, A023360, A077608, A280917, A282906, A282970.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - Sum[Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
  • PARI
    Vec(1/(1 - sum(k=1, 60, (isprime(k) && k%4<3)*x^k)) + O(x^61)) \\ Indranil Ghosh, Mar 15 2017

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A002313(k)).

A309676 Number of compositions (ordered partitions) of n into odd primes (including 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 573, 921, 1481, 2381, 3828, 6153, 9890, 15898, 25556, 41082, 66039, 106156, 170644, 274307, 440945, 708815, 1139412, 1831589, 2944253, 4732847, 7607989, 12229743, 19659153, 31601828, 50799517, 81659549
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2019

Keywords

Links

Crossrefs

Cf. A002124, A006005, A008578, A023360, A080545, A280917, A298603, A298604.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, a(n-1)+
      add(`if`(isprime(j), a(n-j), 0), j=3..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 12 2019
  • Mathematica
    nmax = 42; CoefficientList[Series[1/(1 - x - Sum[x^Prime[k], {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[PrimeOmega[k] < 2 && OddQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

Formula

G.f.: 1/(1 - x - Sum_{k>=2} x^prime(k)).
Previous Showing 11-17 of 17 results.

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

Content is available under The OEIS End-User License Agreement.