A259788 Greatest prime factor of phi(binomial(2*n,n)).

2, 2, 3, 3, 5, 5, 5, 5, 5, 3, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 7, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 83, 83, 83, 83, 83, 83, 89

Offset: 2

Author

Vladimir Shevelev, Jul 05 2015

Keywords

• nonn

Comments

Conjectures:

(1) 7 is a unique term which is not a Sophie Germain prime (A005384);

(2) A Sophie Germain prime p occurs p times if and only if p=2,3,5 and 11; otherwise, it occurs q-p times, where q is the next Sophie Germain prime > p;

(3) a(n) is the greatest prime factor of p-1 for primes p in the interval (n, 2*n).

All these conjectures follow from the following strengthening of the Bertrand postulate for n>=24: the interval (n, 2*n) contains a safe prime (A005385).

Links

• Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Crossrefs

Cf. A000010, A000984, A005384, A005385, A006530, A066973.

Programs

Mathematica

Map[First[Last[FactorInteger[EulerPhi[Binomial[2#,#]]]]]&,Range[2,100]]