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 1-4 of 4 results.

A283875 Number of partitions of n into twin primes (A001097).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 9, 9, 11, 12, 13, 15, 16, 19, 20, 23, 25, 27, 31, 33, 37, 40, 44, 49, 52, 59, 63, 69, 76, 81, 90, 96, 106, 114, 123, 135, 144, 157, 169, 183, 197, 212, 230, 246, 266, 286, 307, 330, 353, 381, 406, 436, 468, 499, 536, 572, 613, 654, 698, 746, 795, 849, 904, 964
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2017

Keywords

Comments

Conjecture: every number > 7 is the sum of at most 4 twin primes (automatically implies the truth of the first version of the twin prime conjecture). For example: 8 = 5 + 3; 9 = 3 + 3 + 3; 10 = 5 + 5; 11 = 5 + 3 + 3; 12 = 7 + 5, etc.

Examples

			a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].

Links

Crossrefs

Cf. A000607, A001097, A077608, A129363, A283876.

Programs

  • Mathematica
    nmax = 79; CoefficientList[Series[Product[1/(1 - Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    Vec(prod(k=1, 79, 1/(1 - (isprime(k) && (isprime(k - 2) || isprime(k + 2)))*x^k)) + O(x^80)) \\ Indranil Ghosh, Mar 17 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).

A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 0, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 0, 2, 2, 0, 1, 2, 2, 2, 2, 3, 3, 3
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Comments

For n > 0 let b(n) be the inverse Euler transform of a(n). It appears that, if p is the lesser of twin primes, then b(p) = 1 and b(2*p) = -1; otherwise b(n) = 0. - Georg Fischer, Aug 15 2020

Examples

			a(46) = 2 because we have [41, 5] and [29, 17].

Links

Crossrefs

Cf. A000586, A000607, A001097, A001359, A077608, A129363, A283875, A283876, A299197.

Programs

  • Maple
    P:= select(isprime,{seq(i,i=3..201,2)}):
TP:= P intersect map(`-`,P,2):
G:= mul(1+x^p,p=TP):
seq(coeff(G,x,i),i=0..200); # Robert Israel, Dec 15 2024
  • Mathematica
    nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k + 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A001359(k)).

A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2, 0, 0, 1, 1, 3, 1, 1, 0, 0, 2, 3, 2, 0, 0, 1, 2, 3, 2, 1, 0, 0, 2, 3, 3, 0, 0, 0, 1, 4, 3, 1, 0, 0, 2, 3, 3, 2, 0, 0, 1, 4, 4
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

Examples

			a(31) = 2 because we have [31] and [19, 7, 5].

Links

Crossrefs

Cf. A000586, A000607, A001097, A006512, A077608, A129363, A283875, A283876, A299196.

Programs

  • Mathematica
    nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k - 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A006512(k)).

A332033 Number of compositions (ordered partitions) of n into distinct twin primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 12, 6, 12, 26, 18, 26, 19, 4, 19, 52, 18, 52, 24, 54, 24, 74, 144, 98, 25, 76, 145, 100, 258, 102, 150, 104, 156, 124, 396, 146, 282, 148, 396, 890, 510, 890, 403, 198, 403, 940, 636, 988, 642
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2020

Keywords

Examples

			a(15) = 6 because we have [7, 5, 3], [7, 3, 5], [5, 7, 3], [5, 3, 7], [3, 7, 5] and [3, 5, 7].

Links

Crossrefs

Cf. A001097, A077608, A219107, A283875, A283876, A331926, A331981.
Showing 1-4 of 4 results.

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