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A116979
Number of distinct representations of primorials as the sum of two primes.
10
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
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. A002375 (number of decompositions of 2n into unordered sums of two odd primes).
Sequence in context: A037154 A037774 A037662 * A351029 A267802 A229928
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