OFFSET
0,11
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.
LINKS
Eric Weisstein's World of Mathematics, Twin Primes
FORMULA
G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).
EXAMPLE
a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].
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]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 17 2017
STATUS
approved