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

0,4

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

Links

Table of n, a(n) for n=0..12.

Eric Weisstein's World of Mathematics, Primorial.

Index entries for sequences related to Goldbach conjecture

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

Example

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.

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 *)

Crossrefs

Cf. A000040, A002110, A351029, A369000.

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

Sequence in context: A037154 A037774 A037662 * A351029 A267802 A229928

Adjacent sequences: A116976 A116977 A116978 * A116980 A116981 A116982

Keyword

nonn,hard,more

Author

Jonathan Vos Post, Apr 01 2006

Extensions

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

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

Status

approved