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 A128199 a(n) = the number of primes S possible, if S = product of b(k)'s + product of c(k)'s, where the distinct positive integers <= n are partitioned into the two sets {b(k)} and {c(k)}. 2
 1, 1, 1, 2, 1, 3, 1, 3, 2, 0, 0, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 1, 1, 0, 2, 3, 2, 4, 2, 0, 0, 3, 1, 1, 2, 4, 4, 1, 5, 5, 5, 5, 4, 2, 0, 3, 4, 11, 4, 6, 8, 4, 4, 6, 6, 8, 6, 2, 2, 4, 4, 11, 7, 6, 13, 13, 19, 42, 15, 19, 14, 18, 19, 30, 38, 29, 12, 24, 24, 41, 15, 10, 12, 28, 19, 22, 27 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(0)=a(1)=1 because the product over the empty set is defined here as 1. For S to be a prime, the positive integers <= n, except 1 and the primes > n/2, must all be together in either {b(k)} or {c(k)}. If p is a prime where n/2 < p <= n, then it is possible that p is in either product of the S sum, as can 1. LINKS EXAMPLE For n = 5 we have the primes 23 = 1*2*4 + 3*5, 29 = 1*2*3*4 + 5, 43 = 1*2*4*5 + 3, so a(5)=3. MATHEMATICA f[n_] := Block[{d = Divisors[Times @@ Select[Range[n], PrimeQ[ # ] && 2# > n &]]}, Select[Union[d + n!/d], PrimeQ]]; Length /@ Array[f, 100, 0] CROSSREFS Cf. A127165, A127166. Sequence in context: A325704 A104984 A083868 * A249617 A304091 A278801 Adjacent sequences:  A128196 A128197 A128198 * A128200 A128201 A128202 KEYWORD nonn AUTHOR Ray Chandler, Feb 18 2007 STATUS approved

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Last modified July 10 07:00 EDT 2020. Contains 335573 sequences. (Running on oeis4.)