This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A072997 #41 Aug 26 2025 04:57:45
%S A072997 2,3,13,31,89,239,617,1571,4007,10141,25673,64853,163367,412007,
%T A072997 1037759,2614369,6584857,16585291,41764859,105178831,264877933,
%U A072997 667038311,1679809291,4230219377,10652786759,26826453991,67555877849
%N A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.
%C A072997 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
%F A072997 It seems that lim_{n -> oo} a(n+1)/a(n) exists and is > 2.
%F A072997 a(n) = A000040(A072986(n)). - _Amiram Eldar_, Aug 24 2018
%F A072997 Limit_{n -> oo} a(n+1)/a(n) = exp(zeta(2)/exp(gamma)) = 2.518... (A362151). - _Amiram Eldar_, Aug 26 2025
%t A072997 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 *)
%o A072997 (PARI) a(n)=if(n<0,0,s=1; while(prod(i=1,s, prime(i)+1)<prod(i=1,s, prime(i))*n,s++); prime(s));
%Y A072997 Cf. A000040, A002110, A023199, A034386, A034448, A072986, A072997, A362151.
%K A072997 nonn,hard,more,changed
%O A072997 1,1
%A A072997 _Benoit Cloitre_, Aug 14 2002
%E A072997 7 more terms from _Ryan Propper_, Jul 22 2005
%E A072997 a(18)-a(22) added by _Amiram Eldar_, Aug 24 2018 from the data at A072986
%E A072997 a(23)-a(27) from _Keith F. Lynch_, Jan 13 2024