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 A236241 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/8 is an integer with C(2*m, m) + prime(m) prime}|, where C(2*m, m) = (2*m)!/(m!)^2, and phi(.) is Euler's totient function. 10
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 2, 3, 4, 5, 2, 2, 2, 3, 4, 3, 2, 4, 4, 6, 3, 5, 8, 9, 6, 6, 4, 5, 5, 4, 5, 6, 6, 4, 4, 4, 10, 9, 7, 4, 4, 5, 7, 2, 2, 3, 7, 7, 5, 7, 6, 7, 5, 4, 7, 5, 5, 3, 8, 6, 4, 6, 5, 8, 9, 5, 4, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,22 COMMENTS Conjecture: a(n) > 0 for every n = 20, 21, ... . We have verified this for n up to 75000. The conjecture implies that there are infinitely many primes of the form C(2*m, m) + prime(m). See A236245 for primes of the form C(2*m, m) + prime(m). See also A236242 for a list of known numbers m with C(2*m, m) + prime(m) prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 EXAMPLE a(20) = 1 since phi(5) + phi(15)/8 = 4 + 1 = 5 with C(2*5,5) + prime(5) = 252 + 11 = 263 prime. a(330) = 1 since phi(211) + phi(330-211)/8 = 210 + 96/8 = 222 with C(2*222,222) + prime(222) = C(444,222) + 1399 prime. MATHEMATICA p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n, n]+Prime[n]] f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/8 a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A000984, A236242, A236245, A236248, A236249, A236256. Sequence in context: A105689 A187200 A117632 * A127731 A159978 A230546 Adjacent sequences:  A236238 A236239 A236240 * A236242 A236243 A236244 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 20 2014 STATUS approved

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Last modified October 17 22:26 EDT 2019. Contains 328134 sequences. (Running on oeis4.)