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 A239214 a(n) = |{0 < k < n: p(k)*p(n)*(p(n)+1) - 1 is prime}|, where p(.) is the partition function (A000041). 4
 0, 1, 2, 3, 1, 3, 3, 2, 3, 3, 5, 4, 4, 3, 3, 6, 2, 4, 5, 4, 1, 2, 3, 6, 6, 6, 2, 4, 6, 9, 2, 7, 8, 6, 6, 2, 2, 2, 10, 4, 4, 7, 5, 7, 1, 4, 9, 9, 9, 4, 6, 8, 7, 8, 6, 4, 13, 10, 3, 6, 10, 7, 13, 12, 12, 8, 6, 8, 5, 11, 5, 3, 4, 5, 11, 7, 6, 12, 16, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Conjecture: (i) a(n) > 0 for all n > 1. (ii) For each n = 2, 3, ... there is a positive integer k < n with p(k)*p(n)*(p(n)-1) + 1 prime. If n > 2, then p(k)*p(n)*(p(n)-1)-1 is prime for some 0 < k < n. (iii) For any n > 1, there is a positive integer k < n with 2*p(k)*p(n)*A000009(n)*A047967(n) + 1 prime. We have verified that a(n) > 0 for all n = 2, ..., 10^5. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014. EXAMPLE a(2) = 1 since p(1)*p(2)*(p(2)+1) - 1 = 1*2*3 - 1 = 5 is prime. a(5) = 1 since p(3)*p(5)*(p(5)+1) - 1 = 3*7*8 - 1 = 167 is prime. a(21) = 1 since p(10)*p(21)*(p(21)+1) - 1 = 42*792*793 - 1 = 26378351 is prime. a(45) = 1 since p(20)*p(45)*(p(45)+1) - 1 = 627*89134*89135 - 1 = 4981489349429 is prime. MATHEMATICA p[n_]:=PartitionsP[n] f[n_]:=p[n]*(p[n]+1) a[n_]:=Sum[If[PrimeQ[p[k]*f[n]-1], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 80}] CROSSREFS Cf. A000040, A000041, A238457, A238509, A238516, A238393, A239207, A239209. Sequence in context: A230294 A104483 A080717 * A200181 A121062 A045831 Adjacent sequences:  A239211 A239212 A239213 * A239215 A239216 A239217 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 12 2014 STATUS approved

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Last modified July 23 22:20 EDT 2021. Contains 346265 sequences. (Running on oeis4.)