

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



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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: A187200 A352340 A117632 * A127731 A159978 A230546
Adjacent sequences: A236238 A236239 A236240 * A236242 A236243 A236244


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 20 2014


STATUS

approved



