A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 89, 2, 5, 409, 571, 2, 809, 227, 317, 2, 5, 79, 23, 4073, 2, 281, 89, 739, 1427, 727, 19, 19, 2, 281, 11, 2143, 2, 1013, 4259, 2, 661, 1879, 401, 5, 4099, 1579, 137, 43, 487, 307, 547, 1709, 43, 3, 463, 2161, 353, 443, 2

Offset: 1

Author

Alexander Adamchuk, Oct 04 2006, Aug 05 2008

Keywords

• nonn

Comments

a(1) = 0 because such a prime does not exist, Mod[n^2+1,n+1] = 2 for n>1.

Corresponding primes (q^p+1)/(q+1), where prime q = a(n) and p = Prime[n], are listed in A123628[n] = {1,3,11,43,683,2731,43691,174763,2796203,402488219476647465854701,715827883,...}.

a(n) coincides with A103795[n] when A103795[n] is prime.

a(n) = 2 for n = PrimePi[A000978[k]] = {2,3,4,5,6,7,8,9,11,14,18,22,26,31,39,43,46,65,69,126,267,380,495,762,1285,1304,1364,1479,1697,4469,8135,9193,11065,11902,12923,13103,23396,23642,31850,...}.

Corresponding primes of the form (2^p + 1)/3 are the Wagstaff primes that are listed in A000979[n] = {3,11,43,683,2731,43691,174763,2796203,715827883,...}.

Links

• Robert Israel, Table of n, a(n) for n = 1..240

Crossrefs

Cf. A123628, A103795, A123487, A123488, A000978, A000979.

Programs

Maple

f:= proc(n) local p,q;
  p:= ithprime(n);
  q:= 1;
  do
   q:= nextprime(q);
   if isprime((q^p+1)/(q+1)) then return q fi
  od
end proc:
f(1):= 0:
map(f, [$1..70]); # Robert Israel, Jul 31 2019

Mathematica

a(1) = 0, for n>1 Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[q], {k, 2, 61}]
Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[{k,q}], {k, 1, 134}]
spq[n_]:=Module[{p=Prime[n],q=2},While[!PrimeQ[(q^p+1)/(q+1)],q=NextPrime[ q]]; q]; Join[{0},Array[spq,70,2]] (* Harvey P. Dale, Mar 23 2019 *)

Formula

A123628(n) = (a(n)^prime(n) + 1) / (a(n) + 1).