A116979 - OEIS

%I #15 Jan 19 2024 16:57:12

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

%T 30309948241

%N Number of distinct representations of primorials as the sum of two primes.

%C 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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial.</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

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

%F a(n) =  A351029(n) - A369000(n). - _Antti Karttunen_, Jan 17 2024

%e a(2) = 1 because 2nd primorial = 6 = 3 + 3 uniquely.

%e a(3) = 3 because 3rd primorial = 30 = 7 + 23 = 11 + 19 = 13 + 17.

%e 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.

%t 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 *)

%Y Cf. A000040, A002110, A351029, A369000.

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

%K nonn,hard,more

%O 0,4

%A _Jonathan Vos Post_, Apr 01 2006

%E More terms from _T. D. Noe_, Apr 28 2006

%E a(11)-a(12) from _Donovan Johnson_, Dec 19 2009