A284721 Smallest odd prime that is relatively prime to 2n+1.

3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 11, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5

Offset: 0

Author

N. J. A. Sloane, Apr 04 2017

Keywords

• nonn

Comments

More than the usual number of terms are shown in order to distinguish this from A239278.

a(n) = smallest odd prime missing from rad(2*n+1).

If rad(2m+1) = rad(2n+1), a(m) = a(n) (cf. A007947). - Bob Selcoe, Apr 04 2017

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Similar to but different from A239278.

Cf. A002110, A007947, A284722, A284723.

Programs

Maple

f:= proc(n) local p;
 p:= 2;
 do
   p:= nextprime(p);
   if igcd(p,2*n+1)=1 then return p fi
 od
end proc:
map(f, [$0..100]); # Robert Israel, Dec 09 2024

Mathematica

a[n_] := Module[{p = 3}, While[Divisible[2*n + 1, p], p = NextPrime[p]]; p]; Array[a, 100, 0] (* Amiram Eldar, Dec 09 2023 *)

PARI

a(n) = my(p=3); while(gcd(2*n+1, p) != 1, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 04 2017

Formula

a(n) = 3 unless n == 1 (mod 3).

For all n >= 2, a(n) < 3*log(2*n+1). Also, for all n >= 1, a(n) < 5*log(2*n+1). [Upper bound corrected by N. J. A. Sloane, Apr 15 2017. Thanks to Bob Selcoe for pointing out that the old bound failed at n=1.]

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{k>=2} (prime(k) * (1/prime(k-1)# - 1/prime(k)#)) = 3.84010195463226942418..., where prime(k)# = A002110(k). - Amiram Eldar, Dec 09 2023