A083341 Smaller factor of the n-th semiprime of the form (m!)^2 + 1.

13, 101, 17, 101, 1344169, 149, 9049, 37, 710341, 2122590346576634509, 171707860473207588349837, 7686544942807799800864250520468090636146175134909, 2196283505473, 598350346949, 1211221552894876996541369232623365900407018851538797

Offset: 1

Author

Hugo Pfoertner, Apr 25 2003

Keywords

• nonn
• hard

Examples

			a(1) = 13 because (A083340(1)!)^2 + 1 = 518401 = 13*39877.
a(15) = 1211221552894876996541369232623365900407018851538797 because (A083340(15)!)^2 + 1 = (55!)^2 + 1 can be factored into P52*P96 with a(15) = P52.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..17
• Andrew Walker, Table of factors of (n!)^2+1.

Crossrefs

Cf. A020549, A083340, subsequence of A282706.

Programs

PARI

for(n=1,29,my(f=(n!)^2+1);if(bigomega(f)==2,print1(vecmin(factor(f)[,1]),", "))) \\ Hugo Pfoertner, Jul 13 2019

Formula

Numbers p such that p*q = (A083340(n)!)^2 + 1, p, q prime, p < q.

Extensions

The 11th term of the sequence (49-digit factor of the 100-digit number (41!)^2+1) was found with Yuji Kida's multiple polynomial quadratic sieve UBASIC PPMPQS v3.5 in 13 days CPU time on an Intel PIII 550 MHz.

Missing a(4) and new a(14), a(15) added by Hugo Pfoertner, Jul 13 2019