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 A234344 a(n) = |{0 < k < n: 2^{phi(k)/2} + 3^{phi(n-k)/2} is prime}|, where phi(.) is Euler's totient function. 14
 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 6, 8, 7, 9, 12, 12, 10, 10, 10, 10, 16, 7, 11, 9, 6, 14, 11, 17, 12, 15, 15, 17, 16, 15, 19, 18, 12, 13, 9, 20, 11, 8, 17, 19, 19, 12, 17, 14, 16, 9, 21, 16, 13, 12, 16, 19, 17, 11, 21, 15, 16, 15, 17, 19, 16, 23, 11, 20, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: a(n) > 0 for all n > 5. This implies that there are infinitely many primes of the form 2^k + 3^m, where k and m are positive integers. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..7000 EXAMPLE a(6) = 1 since 2^{phi(3)/2} + 3^{phi(3)/2} = 5 is prime. a(8) = 3 since 2^{phi(3)/2} + 3^{phi(5)/2} = 11, 2^{phi(4)/2} + 3^{phi(4)/2} = 5, and 2^{phi(5)/2} + 3^{phi(3)/2} = 7 are all prime. MATHEMATICA f[n_, k_]:=2^(EulerPhi[k]/2)+3^(EulerPhi[n-k]/2) a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A000079, A000244, A004051, A234309, A234310, A234337, A234346, A234347 Sequence in context: A097377 A234741 A063917 * A331298 A325351 A279319 Adjacent sequences:  A234341 A234342 A234343 * A234345 A234346 A234347 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 23 2013 STATUS approved

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Last modified July 25 21:44 EDT 2021. Contains 346294 sequences. (Running on oeis4.)