A061232 Number of primes p with n! < p <= (n+1)!.

0, 1, 2, 6, 21, 98, 547, 3556, 26738, 227720, 2170267, 22877331, 264314464, 3320870054, 45076422125, 657316885209, 10247614197601, 170081414212020, 2994059471570761, 55718507205774017, 1092932100469356250, 22536709415953547880, 487361620197926253365

Offset: 0

Author

Amarnath Murthy, Apr 23 2001

Keywords

• nonn
• hard

Comments

First differences of A003604. - Artur Jasinski, Dec 13 2007

Examples

			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.
		

Links

• Andrew R. Booker, The Nth Prime Page

Crossrefs

Cf. A003604.

Programs

Mathematica

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

Formula

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

Extensions

Extended from a(6) on by Patrick De Geest, May 29 2001, using A. Booker's 'Nth Prime Page'

a(15) from Robert G. Wilson v, Jan 29 2003

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

a(17)-a(18) from Donovan Johnson, Oct 30 2012

a(19)-a(22) from A003604(n+1) - A003604(n) by Jinyuan Wang, Mar 11 2020