

A236241


a(n) = {0 < k < n: m = phi(k) + phi(nk)/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
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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

ZhiWei 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(330211)/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[nk]/8
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n1}]
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

ZhiWei Sun, Jan 20 2014


STATUS

approved



