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A234514
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Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 and q(p) + 1 are both prime, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).
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16
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0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 1, 0, 2, 2, 2, 3, 4, 2, 4, 2, 3, 3, 3, 2, 2, 3, 1, 4, 2, 1, 4, 2, 4, 2, 5, 3, 4, 1, 5, 6, 4, 2, 5, 5, 5, 3, 5, 4, 6, 3, 5, 7, 10, 2, 4, 5, 6, 5, 5, 2, 3, 5, 6, 6, 4, 2, 5, 3, 7, 4, 5, 3, 8, 7, 2, 5, 9, 3, 3, 2, 9, 9, 6, 6, 7, 6, 9, 4, 7, 4, 10, 8, 6, 11, 11, 4, 6, 4, 9, 7
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OFFSET
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1,5
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COMMENTS
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Conjecture: (i) a(n) > 0 for all n > 12.
(ii) For any integer n > 4, there is a prime p < n - 2 such that q(p + phi(n-p)/2) + 1 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with q(p) + 1 prime (cf. A234530).
We have verified part (i) for n up to 10^5.
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LINKS
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EXAMPLE
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a(11) = 1 since 11 = 1 + 10 with 1 + phi(10)/2 = 3 and q(3) + 1 = 3 both prime.
a(27) = 1 since 27 = 7 + 20 with 7 + phi(20)/2 = 11 and q(11) + 1 = 13 both prime.
a(30) = 1 since 30 = 8 + 22 with 8 + phi(22)/2 = 13 and q(13) + 1 = 19 both prime.
a(38) = 1 since 38 = 21 + 17 with 21 + phi(17)/2 = 29 and q(29) + 1 = 257 both prime.
a(572) = 1 since 572 = 77 + 495 with 77 + phi(495)/2 = 197 and q(197) + 1 = 406072423 both prime.
a(860) = 1 since 860 = 523 + 337 with 523 + phi(337)/2 = 691 and q(691) + 1 = 712827068077888961 both prime.
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MATHEMATICA
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f[n_, k_]:=k+EulerPhi[n-k]/2
q[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[PartitionsQ[f[n, k]]+1]
a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
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CROSSREFS
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Cf. A000009, A000010, A000040, A229835, A233307, A233390, A233417, A234451, A234470, A234475, A234503, A234504, A234530
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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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