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

Author

Ilya Gutkovskiy, Feb 04 2018

Keywords

• nonn

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

• 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

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)).