login
Number of primes p with n! < p <= (n+1)!.
5

%I #22 Apr 03 2023 10:36:10

%S 0,1,2,6,21,98,547,3556,26738,227720,2170267,22877331,264314464,

%T 3320870054,45076422125,657316885209,10247614197601,170081414212020,

%U 2994059471570761,55718507205774017,1092932100469356250,22536709415953547880,487361620197926253365

%N Number of primes p with n! < p <= (n+1)!.

%C First differences of A003604. - _Artur Jasinski_, Dec 13 2007

%H Andrew R. Booker, <a href="https://t5k.org/nthprime/">The Nth Prime Page</a>

%F I conjecture that for n>2 we have n + 1/2 <= a(n)/a(n-1) <= n + 2/3. If this conjecture is true we have floor(a(n+1)/a(n)) = n. - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 03 2006

%e a(3) = 6 as there are 6 primes between 3! = 6 and 4! = 24: 7,11,13,17,19,23; a(4) = 21 as there are 21 primes between 24 and 120.

%t Table[PrimePi[(n + 1)! ] - PrimePi[n! ], {n, 0, 15}]

%Y Cf. A003604.

%K nonn,hard

%O 0,3

%A _Amarnath Murthy_, Apr 23 2001

%E Extended from a(6) on by _Patrick De Geest_, May 29 2001, using A. Booker's 'Nth Prime Page'

%E a(15) from _Robert G. Wilson v_, Jan 29 2003

%E Edited by _N. J. A. Sloane_, May 15 2008 at the suggestion of _R. J. Mathar_

%E a(17)-a(18) from _Donovan Johnson_, Oct 30 2012

%E a(19)-a(22) from A003604(n+1) - A003604(n) by _Jinyuan Wang_, Mar 11 2020