A104189 Prime numbers arising from Schorn's proof that there are infinitely many primes.

2, 3, 5, 7, 13, 19, 73, 97, 241, 601, 2161, 15121, 20161, 30241, 35281, 161281, 241921, 282241, 1088641, 1451521, 2177281, 2903041, 10886401, 18144001, 29030401, 32659201, 39916801, 199584001, 319334401, 958003201, 2395008001, 2874009601

Offset

1,1

Comments

a(1045) has 1003 digits. - Michael S. Branicky, Apr 30 2025

References

Paolo Ribenboim, "The New Book of Prime Number Records", 1996, ISBN 0-387-94457-5 Page 5

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1044

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From N. J. A. Sloane, Jun 13 2012

Peter Schorn, Schorn's Proof

www.mathematic.de, Schorn's proof

Formula

For each n >= 1, add all n!*i+1, where 1 <= i <= n and n!*i+1 is a prime. [Clarified by Michael S. Branicky, Apr 30 2025.]

Example

6!*3+1 = 2161 is prime and is a term.

Mathematica

lst={}; Do[lst=Join[lst, Select[n!Range[n]+1, PrimeQ]], {n, 12}]; lst (* T. D. Noe, Nov 02 2006 *)

Prog

(Python)
from math import factorial
from gmpy2 import is_prime
from itertools import count, islice
def agen():
    for n in count(1):
        f = factorial(n)
        yield from (t for i in range(1, n+1) if is_prime(t:=f*i+1))
print(list(islice(agen(), 32))) # Michael S. Branicky, Apr 30 2025

Crossrefs

Sequence in context: A147485 A341650 A341640 * A301776 A380026 A178570

Adjacent sequences: A104186 A104187 A104188 * A104190 A104191 A104192

Keyword

nonn

Author

Karsten Meyer, Mar 12 2005; extended Jun 08 2005

Extensions

Corrected by T. D. Noe, Nov 02 2006

Status

approved