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 A235805 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function. 2
 0, 0, 0, 0, 0, 2, 3, 2, 4, 4, 4, 4, 4, 4, 2, 6, 3, 7, 4, 2, 7, 3, 5, 4, 4, 6, 6, 6, 4, 4, 7, 8, 9, 6, 6, 11, 8, 10, 6, 6, 12, 8, 13, 6, 12, 8, 13, 10, 7, 14, 10, 11, 11, 11, 16, 14, 13, 9, 15, 11, 23, 14, 12, 11, 12, 10, 14, 8, 15, 9, 14, 13, 11, 12, 9, 19, 9, 14, 11, 16, 8, 14, 5, 13, 8, 13, 9, 13, 10, 15, 10, 11, 12, 17, 8, 13, 10, 11, 7, 18 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: a(n) > 0 for all n > 5. This implies that there are infinitely many odd primes p = 2*m + 1 with m*(m+1) - prime(m) and m*(m+1)- prime(m+1) both prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(8) = 2 since phi(4) + phi(4)/2 = 3 with 2*3 + 1 = 7, 3*4 - prime(3) = 7 and 3*4 - prime(4) = 5 all prime, and phi(5) + phi(3)/2 = 5 with 2*5 + 1 = 11, 5*6 - prime(5) = 19 and 5*6 - prime(6) = 17 all prime. MATHEMATICA q[n_]:=PrimeQ[2n+1]&&PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)-Prime[n+1]] f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2 a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n-3}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A235592, A235728, A235806. Sequence in context: A304739 A304735 A318048 * A168231 A235728 A205546 Adjacent sequences:  A235802 A235803 A235804 * A235806 A235807 A235808 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 16 2014 STATUS approved

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Last modified October 23 19:26 EDT 2021. Contains 348215 sequences. (Running on oeis4.)