A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).

2, 3, 7, 43, 13, 3263443, 547, 29881, 5295435634831, 181, 2287, 73

Offset: 0

Author

Robert FERREOL, Jan 19 2019

Keywords

• nonn
• more

Comments

a(n) is also the smallest prime divisor of A007018(n+1) that is not a divisor of A007018(n).

The prime numbers a(n) are all distinct, which proves the infinitude of the prime numbers (Saidak's proof).

a(12) <= 2589377038614498251653. - Daniel Suteu, Jan 20 2019

a(12)..a(50) = [?, 52387, 13999, 17881, 128551, 635263, ?, ?, 352867, 387347773, ?, 74587, ?, ?, 27061, 164299, 20929, 1171, ?, 1679143, ?, ?, 120823, 2408563, 38218903, 333457, 30241, 4219, 1085443, 7603, 1861, ?, 23773, 51769, 1285540933, 429547, ?, 8323570543, ?], where ? denote unknown values > 10^10. - Max Alekseyev, Oct 11 2023

Links

• Filip Saidak, A new proof of Euclid's theorem, Amer. Math. Monthly, 113:10 (2006) 937-938.
• Wikipedia, Sylvester's sequence: Divisibility and factorizations.

Crossrefs

Cf. A007018, A000058, A007996, A091335, A180871.

Programs

Maple

with(numtheory):
u:=1: P:=NULL: to 9 do P:=P,sort([op(divisors(u+1))])[2]: u:=u*(u+1) od:
P;

PARI

f(n)=if(n<1, n>=0, f(n-1)+f(n-1)^2); \\ A007018
a(n)=divisors(f(n)+1)[2]; \\ Michel Marcus, Jan 20 2019

Formula

a(n) = A007996(m), where m is the smallest index such that A180871(m) = n. - Max Alekseyev, Oct 11 2023

Extensions

a(10)-a(11) from Daniel Suteu, Jan 20 2019