A116979 Number of distinct representations of primorials as the sum of two primes.

0, 0, 1, 3, 19, 114, 905, 9493, 124180, 2044847, 43755729, 1043468386, 30309948241

Offset: 0

Author

Jonathan Vos Post, Apr 01 2006

Keywords

• nonn
• hard
• more

Comments

Related to Goldbach's conjecture. Let g(2n) = A002375(n). The primorials produce maximal values of the function g in the following sense: the basic shape of the function g is k*x/log(x)^2 and each primorial requires a larger value of k than the previous one. - T. D. Noe, Apr 28 2006

Relates also to a more generic problem of how many numbers there are such that their arithmetic derivative is equal to the n-th primorial number. See A351029. - Antti Karttunen, Jan 17 2024

Examples

			a(2) = 1 because 2nd primorial = 6 = 3 + 3 uniquely.
a(3) = 3 because 3rd primorial = 30 = 7 + 23 = 11 + 19 = 13 + 17.
a(4) = 19 because 4th primorial = 210 = 11 + 199 = 13 + 197 = 17 + 193 = 19 + 191 = 29 + 181 = 31 + 179 = 37 + 173 = 43 + 167 = 47 + 163 = 53 + 157 = 59 + 151 = 61 + 149 = 71 + 139 = 73 + 137 = 79 + 131 = 83 + 127 = 97 + 113 = 101 + 109 = 103 + 107.
		

Links

• Eric Weisstein's World of Mathematics, Primorial.
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A000040, A002110, A351029, A369000.

Cf. A002375 (number of decompositions of 2n into unordered sums of two odd primes).

Programs

Mathematica

n=1; Join[{0,0}, Table[n=n*Prime[k]; cnt=0; Do[If[PrimeQ[2n-Prime[i]],cnt++ ], {i,2,PrimePi[n]}]; cnt, {k,2,10}]] (* T. D. Noe, Apr 28 2006 *)

Formula

a(n) = #{p(i) + p(j) = A002110(n) for p(k) = A000040(k) and i >= j}.

a(n) = A351029(n) - A369000(n). - Antti Karttunen, Jan 17 2024

Extensions

More terms from T. D. Noe, Apr 28 2006

a(11)-a(12) from Donovan Johnson, Dec 19 2009