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

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

0,47

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

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Ilya Gutkovskiy, Scatter plot of a(n) - A299197(n)

Eric Weisstein's World of Mathematics, Twin Primes

Index entries for related partition-counting sequences

Formula

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

Example

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

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]

Crossrefs

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

Sequence in context: A114525 A127672 A294168 * A227698 A331466 A166124

Adjacent sequences: A299193 A299194 A299195 * A299197 A299198 A299199

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 04 2018

Status

approved