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 A237130 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that both {3*k - 1, 3*k + 1} and {phi(m) - 1, phi(m) + 1} are twin prime pairs, where phi(.) is Euler's totient function. 7
 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 4, 3, 3, 1, 3, 4, 4, 3, 3, 5, 5, 3, 2, 2, 3, 4, 2, 3, 5, 5, 3, 4, 4, 5, 3, 5, 2, 3, 4, 4, 4, 2, 6, 4, 3, 4, 3, 5, 1, 5, 5, 5, 4, 2, 5, 4, 4, 2, 4, 6, 5, 6, 3, 5, 5, 6, 5, 1, 5, 3, 5, 3, 6, 4, 5, 7, 3, 5, 3, 5, 5, 3, 7, 3, 9, 4, 6, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS Conjecture: (i) a(n) > 0 for all n > 8. (ii) Any integer n > 6 can be written as k + m with k > 0 and m > 0 such that both {prime(k), prime(k) + 2} and {phi(m) - 1, phi(m) + 1} are twin prime pairs. (iii) Each n = 12, 13, ... can be written as p + q (q > 0) with p, p + 6, phi(q) - 1 and phi(q) + 1 all prime. (iv) If n > 2 is neither 10 nor 430, then n can be written as k + m with k > 0 and m > 0 such that both {3k - 1, 3*k + 1} and {6*m - 1, 6*m + 1} are twin prime pairs. Note that each part of the above conjecture implies the twin prime conjecture. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(7) = 1 since 7 = 2 + 5 with 3*2 - 1 = 5, 3*2 + 1 =7, phi(5) - 1 = 3 and phi(5) + 1 = 5 all prime. a(140) = 1 since 140 = 104 + 36 with 3*104 - 1 = 311, 3*104 + 1 = 313, phi(36) - 1 = 11 and phi(36) + 1 = 13 all prime. MATHEMATICA PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1] a[n_]:=Sum[If[PrimeQ[3k-1]&&PrimeQ[3k+1]&&PQ[n-k], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A001359, A006512, A014574, A072281, A182662, A236531, A236566, A236831, A236968, A237127. Sequence in context: A327983 A205784 A066272 * A330524 A336037 A058773 Adjacent sequences:  A237127 A237128 A237129 * A237131 A237132 A237133 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 04 2014 STATUS approved

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Last modified September 18 23:30 EDT 2020. Contains 337175 sequences. (Running on oeis4.)