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 A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function. 7
 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 0, 5, 5, 2, 4, 1, 5, 3, 3, 2, 4, 4, 9, 5, 9, 4, 10, 3, 6, 6, 8, 5, 10, 4, 4, 7, 8, 10, 5, 8, 9, 9, 4, 11, 3, 5, 5, 9, 5, 4, 4, 5, 6, 8, 7, 6, 3, 11, 4, 8, 10, 9, 8, 7, 6, 11, 7, 9, 4, 6, 5, 6, 2, 9, 4, 7, 6, 7, 10, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Conjecture: a(n) > 0 for all n > 31. This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 EXAMPLE a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime. a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime. MATHEMATICA p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1] f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1 a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-3}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119. Sequence in context: A264033 A236293 A056044 * A239319 A236468 A116685 Adjacent sequences:  A236094 A236095 A236096 * A236098 A236099 A236100 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 19 2014 STATUS approved

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Last modified January 26 18:22 EST 2021. Contains 340442 sequences. (Running on oeis4.)