A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.

2, 3, 13, 31, 89, 239, 617, 1571, 4007, 10141, 25673, 64853, 163367, 412007, 1037759, 2614369, 6584857, 16585291, 41764859, 105178831, 264877933, 667038311, 1679809291, 4230219377, 10652786759, 26826453991, 67555877849

Offset: 1

Author

Benoit Cloitre, Aug 14 2002

Keywords

• nonn
• hard
• more
• changed

Comments

For k > 2, the primorial number A034386(A072997(k)) = A002110(A072986(k)) is the least unitary k-abundant number, i.e., the least number m such that usigma(m) >= k*m, where usigma(m) = A034448(m) is the sum of unitary divisors of m. The sequence of these primorials is the unitary version of A023199. - Amiram Eldar, Aug 24 2018

Crossrefs

Cf. A000040, A002110, A023199, A034386, A034448, A072986, A072997, A362151.

Programs

Mathematica

n=x=y=1; Do[x *= (Prime[s] + 1); y *= Prime[s]; If[x >= n*y, Print[Prime[s]]; n++ ], {s, 1, 10^6}] (* Ryan Propper, Jul 22 2005 *)

PARI

a(n)=if(n<0,0,s=1; while(prod(i=1,s, prime(i)+1)
				

Formula

It seems that lim_{n -> oo} a(n+1)/a(n) exists and is > 2.

a(n) = A000040(A072986(n)). - Amiram Eldar, Aug 24 2018

Limit_{n -> oo} a(n+1)/a(n) = exp(zeta(2)/exp(gamma)) = 2.518... (A362151). - Amiram Eldar, Aug 26 2025

Extensions

7 more terms from Ryan Propper, Jul 22 2005

a(18)-a(22) added by Amiram Eldar, Aug 24 2018 from the data at A072986

a(23)-a(27) from Keith F. Lynch, Jan 13 2024