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A238646
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Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.
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2
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0, 0, 0, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 1, 3, 1, 4, 2, 3, 2, 5, 4, 5, 1, 3, 3, 4, 2, 5, 3, 4, 5, 8, 3, 5, 1, 5, 5, 7, 3, 5, 2, 6, 3, 6, 6, 9, 4, 8, 7, 7, 6, 7, 4, 6, 7, 8, 5, 6, 4, 7, 8, 9, 6, 6, 6, 9, 5, 7, 4, 8, 6, 10, 6, 5, 8, 11, 7, 10, 6
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OFFSET
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1,5
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COMMENTS
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Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 10, 12, 14, 16, 24, 36.
This is analog of the conjecture in A237705 for squarefree numbers.
We have verified the conjecture for n up to 60000.
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LINKS
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EXAMPLE
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a(10) = 1 since 7 and 3 are both prime, and there are exactly 3 squarefree numbers among 1, ..., 10-7.
a(36) = 1 since 17 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., 36-17 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).
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MATHEMATICA
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s[n_]:=Sum[If[SquareFreeQ[k], 1, 0], {k, 1, n}]
a[n_]:=Sum[If[PrimeQ[s[n-Prime[k]]], 1, 0], {k, 1, PrimePi[n-1]}]
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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