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 21-26 of 26 results.

A106245 Number of partitions into powers of distinct primes.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 5, 5, 5, 6, 7, 7, 8, 8, 12, 12, 12, 14, 15, 17, 18, 20, 23, 25, 25, 29, 32, 35, 36, 38, 44, 46, 48, 54, 60, 62, 62, 68, 77, 82, 86, 92, 97, 103, 109, 119, 131, 135, 138, 147, 161, 174, 182, 191, 201, 210, 223, 242, 260, 268, 273, 288, 315, 336, 354
Offset: 1

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Author

Reinhard Zumkeller, Apr 26 2005

Keywords

Examples

			a(12) = #{11+1,3^2+2+1,2^3+3+1,7+5,7+2^2+1,7+3+2,5+2^2+3} =
7.

Crossrefs

Cf. A106244, A051613, A000961.

Formula

a(n) = A051613(n-1)+A051613(n). - Vladeta Jovovic, Apr 28 2005

A280126 Expansion of Product_{k>=1} (1 + x^(prime(k)^2))*(1 + x^(prime(k)^3)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2016

Keywords

Comments

Number of partitions of n into distinct parts that are squares of primes (A001248) or cubes of primes (A030078).

Examples

			a(61) = 2 because we have [49, 8, 4] and [25, 27, 9].

Links

Crossrefs

Cf. A000586, A001248, A030078, A106244, A111900, A111902, A131799, A168363, A280125.

Programs

  • Mathematica
    nmax = 120; CoefficientList[Series[Product[(1 + x^Prime[k]^2) (1 + x^Prime[k]^3), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(prime(k)^2))*(1 + x^(prime(k)^3)).

A331925 Number of compositions (ordered partitions) of n into distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 11, 17, 19, 48, 49, 62, 85, 120, 258, 175, 337, 464, 631, 646, 932, 1686, 1991, 2122, 2455, 4118, 4545, 6010, 6481, 13302, 14383, 16177, 16912, 26454, 32024, 35468, 42389, 57334, 107708, 73830, 125629, 142560, 200377, 172752, 244624
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(6) = 10 because we have [5, 1], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Links

Crossrefs

Cf. A000961, A023893, A106244, A219107, A280543, A331847.

Programs

  • Maple
    N:= 50: # for a(0)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]):
PP:= sort([1,seq(seq(p^j, j = 1 .. ilog[p](N)),p=P)]):G:= 1:
for s in PP do
  G:= G + series(G*x*y^s,y,N+1);
od:
G:= convert(G,polynom):
T:= add(coeff(G,x,i)*i!,i=0..N):
seq(coeff(T,y,i),i=0..N); # Robert Israel, Jun 28 2024

A352166 Number of partitions of n into distinct odd prime powers (1 included).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 12, 12, 13, 14, 16, 17, 17, 19, 21, 23, 23, 25, 28, 30, 31, 33, 37, 38, 40, 43, 47, 50, 52, 55, 60, 64, 66, 70, 76, 81, 83, 89, 96, 101, 105, 110, 119, 125, 130, 138, 147, 155, 161
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2022

Keywords

Crossrefs

Cf. A000961, A024939, A054685, A061345, A106244, A134337, A280152, A352165.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[(1 + Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=0} (1 + x^A061345(k)).

A356067 Number of integer partitions of n into relatively prime prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 11, 7, 18, 16, 26, 27, 43, 41, 65, 65, 92, 100, 137, 142, 194, 210, 270, 295, 379, 410, 519, 571, 699, 782, 947, 1046, 1267, 1414, 1673, 1870, 2213, 2465, 2897, 3230, 3757, 4210, 4871, 5427, 6265, 6997
Offset: 0

Views

Author

Gus Wiseman, Jul 28 2022

Keywords

Examples

			The a(5) = 1 through a(12) = 7 partitions:
  (32)  .  (43)   (53)   (54)    (73)    (74)     (75)
           (52)   (332)  (72)    (433)   (83)     (543)
           (322)         (432)   (532)   (92)     (552)
                         (522)   (3322)  (443)    (732)
                         (3222)          (533)    (4332)
                                         (542)    (5322)
                                         (722)    (33222)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Crossrefs

This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.
For strict instead of coprime: A054685, facs A050361, with 1's A106244.
The version for factorizations instead of partitions is A354911.
A000041 counts partitions, strict A000009.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A279784 counts twice-partitions where the latter partitions are constant.
A289509 lists numbers whose prime indices are relatively prime.
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&&GCD@@#==1&]],{n,0,30}]

A280596 Expansion of Product_{p prime, k>=2} (1 + x^(p^k)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 3, 1, 4, 4, 3, 1, 4, 5, 3, 2, 4, 6, 3, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 7, 4, 2, 4, 7, 5, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 06 2017

Keywords

Comments

Number of partitions of n into distinct proper prime powers (A246547).

Examples

			a(25) = 2 because we have [25] and [16, 9].

Links

Crossrefs

Cf. A054685, A106244, A246547, A280586.

Programs

  • Mathematica
    nmax = 107; CoefficientList[Series[Product[(1 + Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{p prime, k>=2} (1 + x^(p^k)).
Previous Showing 21-26 of 26 results.

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