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A234360
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a(n) = |{0 < k < n: (k+1)^{phi(n-k)} + k is prime}|, where phi(.) is Euler's totient function.
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10
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0, 1, 2, 3, 3, 4, 6, 4, 4, 7, 6, 5, 9, 5, 5, 9, 8, 9, 6, 5, 9, 7, 8, 9, 6, 8, 7, 4, 7, 8, 12, 8, 6, 7, 8, 7, 11, 5, 6, 11, 7, 10, 5, 9, 4, 10, 9, 7, 8, 9, 8, 8, 8, 9, 7, 7, 5, 10, 7, 3, 12, 5, 7, 7, 9, 8, 8, 5, 14, 6, 9, 4, 10, 2, 7, 7, 8, 2, 7, 9, 10, 7, 8, 5, 7
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OFFSET
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1,3
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 1. Also, for any n > 5 there is a positive integer k < n with (k+1)^{phi(n-k)/2} - k prime.
(ii) If n > 1, then k*(k+1)^{phi(n-k)} + 1 is prime for some 0 < k < n. If n > 3, then k*(k+1)^{phi(n-k)/2} - 1 is prime for some 0 < k < n.
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LINKS
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EXAMPLE
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a(74) = 2 since (2+1)^{phi(72)} + 2 = 3^{24} + 2 =
282429536483 and (14+1)^{phi(60)} + 14 = 15^{16} + 14 = 6568408355712890639 are both prime.
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MATHEMATICA
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f[n_, k_]:=f[n, k]=(k+1)^(EulerPhi[n-k])+k
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
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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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