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A238577
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a(n) = |{0 < k <= n: p(n)*q(k)*r(k) + 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.
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3
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0, 1, 1, 2, 1, 2, 4, 3, 4, 3, 7, 4, 5, 6, 4, 4, 6, 4, 7, 1, 4, 6, 2, 8, 6, 6, 5, 4, 5, 4, 8, 5, 9, 3, 4, 2, 3, 10, 5, 11, 5, 10, 5, 6, 3, 6, 8, 7, 9, 6, 6, 3, 10, 3, 9, 9, 6, 10, 8, 8, 7, 4, 6, 6, 6, 5, 3, 9, 4, 8, 12, 5, 2, 8, 8, 3, 6, 10, 9, 9
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OFFSET
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1,4
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 5, 20. If n > 2, then p(n)*q(k)*r(k) - 1 is prime for some k = 1, ..., n.
(ii) If n > 2 is not equal to 22, then p(n)*q(n)*q(k) - 1 is prime for some k = 1, ..., n. If n > 13, then p(n)*q(k)*q(n-k) - 1 is prime for some 1 < k < n/2.
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LINKS
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EXAMPLE
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a(5) = 1 since p(5)*q(4)*r(4) + 1 = 7*2*3 + 1 = 43 is prime.
a(20) = 1 since p(20)*q(13)*r(13) + 1 = 627*18*83 + 1 = 936739 is prime.
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MATHEMATICA
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p[n_, k_]:=PrimeQ[PartitionsP[n]*PartitionsQ[k]*(PartitionsP[k]-PartitionsQ[k])+1]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 80}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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