A100395 The smallest prime number q such that the greatest prime divisor of 2*q+1 equals the n-th prime.

13, 2, 3, 5, 19, 59, 47, 11, 43, 139, 277, 61, 107, 23, 79, 29, 457, 167, 461, 109, 197, 41, 311, 727, 151, 257, 53, 163, 2203, 317, 1637, 479, 347, 223, 1283, 863, 733, 83, 1297, 89, 271, 859, 1061, 1871, 2089, 5591, 557, 113, 1259, 349, 1553, 3253, 1129, 2441

Offset: 2

Author

Labos Elemer, Dec 16 2004

Keywords

• nonn

Comments

The offset is 2 because prime(1)=2 is never a prime factor of an odd number.

Examples

			n=1: a(1)=13 because it is the least prime number such that the greatest prime divisor of 2*13 + 1 = 27 equals 3;
n=2: a(2)=2 because the largest prime divisor of 2*a(2) + 1 = 5 is 5;
n=6: a(6)=19 since the greatest prime factor of 2*19 + 1 = 39 = 3*13 is 13=prime(6).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10001

Crossrefs

Cf. A006530, A023590, A100394.

Programs

Maple

A100395 := proc(n)
    p := ithprime(n) ;
    for i from 1 do
        q := ithprime(i) ;
        numtheory[factorset](2*q+1) ;
        if max(op(%)) = p then
            return q;
        end if;
    end do:
end proc:
seq(A100395(n),n=2..60) ; # R. J. Mathar, Sep 22 2018

Mathematica

gpf[n_] := FactorInteger[n][[-1, 1]]; n = 54; m = Prime[n + 1]; v = Table[0, {m}]; c = 0; p = 2; While[c < n, g = gpf[2*p + 1]; If[g <= m && v[[g]] == 0, c++; v[[g]] = p]; p = NextPrime[p]]; Select[v, # > 0 &] (* Amiram Eldar, Aug 08 2020 *)

Formula

a(n) = Min{x; x is prime number; A006530(2x+1) = prime(n)}.