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 A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function. 5
 0, 0, 0, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 6, 3, 6, 0, 6, 4, 5, 3, 1, 3, 4, 2, 3, 4, 1, 8, 6, 4, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n) > 0 for all n > 84. Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime. a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime. a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime. MATHEMATICA PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]] f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1 a[n_]:=Sum[If[PQ[f[n, k]], 1, 0], {k, 1, n-3}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A232353, A234694, A234695, A235189, A235330, A235613, A235614, A235661, A235681. Sequence in context: A152538 A345033 A329359 * A324830 A141110 A325758 Adjacent sequences:  A235679 A235680 A235681 * A235683 A235684 A235685 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 13 2014 STATUS approved

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Last modified July 30 09:18 EDT 2021. Contains 346359 sequences. (Running on oeis4.)