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 A234337 a(n) = |{0 < k < n - 2: 4^k + 2^{phi(n-k)} - 1 is prime}|, where phi(.) is Euler's totient function. 14
 0, 0, 0, 1, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 8, 7, 6, 8, 6, 10, 8, 5, 6, 7, 10, 7, 6, 10, 9, 6, 7, 8, 12, 5, 9, 4, 9, 4, 6, 3, 8, 8, 11, 10, 9, 7, 7, 13, 12, 6, 7, 8, 6, 6, 13, 10, 8, 9, 9, 12, 6, 11, 14, 9, 5, 11, 7, 7, 10, 11, 7, 9, 10, 5, 9, 8, 8, 13, 7, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: Let a be 2 or 3 or 4. If n > 3, then a^k + a^{phi(n-k)/2} - 1 is prime for some 0 < k < n - 2. This conjecture for a = 4 implies that there are infinitely many terms of the sequence A234310. The conjecture for a = 3 implies that there are infinitely many primes of the form 3^k + 3^m - 1 (cf. A234346), where k and m are positive integers. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..4000 EXAMPLE a(4) = 1 since 4^1 + 2^{phi(3)} - 1 = 7 is prime. a(5) = 2 since 4^1 + 2^{phi(4)} - 1 = 7 and 4^2 + 2^{phi(3)} - 1 = 19 are both prime. MATHEMATICA f[n_, k_]:=4^k+2^(EulerPhi[n-k])-1 a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-3}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A000244, A000302, A234309, A234310, A234344, A234346, A234347, A234359, A234360, A234361 Sequence in context: A227559 A234968 A141784 * A216391 A014202 A309247 Adjacent sequences:  A234334 A234335 A234336 * A234338 A234339 A234340 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 23 2013 STATUS approved

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)