OFFSET
1,2
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) = 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
FORMULA
A123628(n) = (a(n)^prime(n) + 1) / (a(n) + 1).
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Oct 04 2006, Aug 05 2008
STATUS
approved