A051342 Smallest prime factor of 1 + (product of first n primes).

3, 7, 31, 211, 2311, 59, 19, 347, 317, 331, 200560490131, 181, 61, 167, 953, 73, 277, 223, 54730729297, 1063, 2521, 22093, 265739, 131, 2336993, 960703, 2297, 149, 334507, 5122427, 1543, 1951, 881, 678279959005528882498681487, 87549524399, 23269086799180847

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Based on Euclid's proof that there are infinitely many primes.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..87
• M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• R. G. Wilson v, Explicit factorizations

Crossrefs

Cf. A014545, A002585.

Programs

Maple

a:= proc(n)
local N, F, i;
  N:= 1 + mul(ithprime(i),i=1..n);
  F:= select(type,map(t->t[1],ifactors(N,easy)[2]),integer);
if nops(F) >= 1 then return min(F) fi;
  min(map(t->t[1],ifactors(N)[2]))
end proc:
seq(a(n),n=1..40); # Robert Israel, Oct 19 2014

Mathematica

Map[FactorInteger,
   Table[Product[Prime[n], {n, 1, m}] + 1, {m, 1, 36}]][[All,
1]][[All, 1]] (* Geoffrey Critzer, Oct 19 2014 *)
FactorInteger[#][[1,1]]&/@(FoldList[Times,Prime[Range[40]]]+1) (* Harvey P. Dale, Oct 08 2021 *)

PARI

a(n) = factor(1 + prod(i=1, n, prime(i)))[1, 1]; \\ Michel Marcus, Dec 10 2013

Formula

a(n) = A020639(1+A002110(n)).

Extensions

One more term from Michel Marcus, Dec 10 2013