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Euclid-Pocklington primes: primes of the form Product_{i=1..k} prime(i) * prime(k+1)^m + 1 where prime(r) is the r-th prime and Product_{i=1..k} prime(i) < prime(k+1)^m.
3

%I #24 Jun 25 2021 03:29:05

%S 3,5,7,17,19,151,163,257,487,751,1459,1471,39367,65537,72031,279511,

%T 33820711,86093443,258280327,372027811,4092305911,11149928791,

%U 42638305711,209481995953231,411782264189299,3561193931204911

%N Euclid-Pocklington primes: primes of the form Product_{i=1..k} prime(i) * prime(k+1)^m + 1 where prime(r) is the r-th prime and Product_{i=1..k} prime(i) < prime(k+1)^m.

%C Named after the Greek mathematician Euclid (flourished c. 300 B.C.) and the English physicist and mathematician Henry Cabourn Pocklington (1870-1952). - _Amiram Eldar_, Jun 24 2021

%H Donovan Johnson, <a href="/A053341/b053341.txt">Table of n, a(n) for n = 1..454</a>

%e 5 = 2^2+1 is of this form (with k=0).

%t eucpock[max_] := Module[{p = 1, prod = 1, m, q, r, s = {}}, While[prod < max, prod *= p; q = NextPrime[p]; m = Max[1, Ceiling @ Log[q, prod]]; r = prod * q^m; While[r + 1 <= max, If[PrimeQ[r + 1], AppendTo[s, r + 1]]; r *= q]; p = NextPrime[p]]; Union[s]]; eucpock[10^16] (* _Amiram Eldar_, Jun 24 2021 *)

%Y Cf. A006862.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_, Jan 05 2000

%E More terms from _Jud McCranie_, Jul 11 2000