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 A236265 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! - prime(m) prime}|, where phi(.) is Euler's totient function. 3
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 2, 2, 4, 3, 5, 1, 3, 2, 3, 3, 4, 5, 9, 5, 5, 6, 7, 8, 8, 8, 5, 7, 5, 8, 8, 5, 5, 9, 8, 6, 6, 9, 8, 10, 6, 9, 4, 6, 9, 9, 8, 10, 9, 6, 10, 7, 8, 12, 11, 10, 8, 11, 9, 12, 7, 13, 12, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,22 COMMENTS It seems that a(n) > 0 for all n > 21. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! - prime(m) prime. See also A236263 for a similar sequence. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..7000 EXAMPLE a(23) = 1 since phi(7)/2 + phi(16)/8 = 3 + 1 = 4 with 4! - prime(4) = 24 - 7 = 17 prime. a(26) = 1 since phi(9)/2 + phi(17)/8 = 3 + 2 = 5 with 5! - prime(5) = 120 - 11 = 109 prime. MATHEMATICA q[n_]:=IntegerQ[n]&&PrimeQ[n!-Prime[n]] f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8 a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263. Sequence in context: A280521 A278043 A014643 * A238645 A118382 A007723 Adjacent sequences:  A236262 A236263 A236264 * A236266 A236267 A236268 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 21 2014 STATUS approved

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Last modified June 19 18:24 EDT 2021. Contains 345144 sequences. (Running on oeis4.)