A055211 Lesser Fortunate numbers.

3, 7, 11, 13, 17, 29, 23, 43, 41, 73, 59, 47, 89, 67, 73, 107, 89, 101, 127, 97, 83, 89, 97, 251, 131, 113, 151, 263, 251, 223, 179, 389, 281, 151, 197, 173, 239, 233, 191, 223, 223, 293, 593, 293, 457, 227, 311, 373, 257, 307, 313, 607, 347, 317, 307, 677, 467

Offset: 2

Author

Robert G. Wilson v, Jul 04 2000

Keywords

• nonn

Comments

a(1) is not defined. The first 1000 terms are all prime and it is conjectured that all terms are primes.

a(n) is the smallest m such that m > 1 and A002110(n) - m is prime. For n > 2, a(n) must be greater than prime(n+1) - 1. - Farideh Firoozbakht, Aug 20 2003

Examples

			a(3) = 7 since 2*3*5 = 30, 30-1 = 29, previous prime is 23, 30-23 = 7.
		

Links

• Pierre CAMI, Table of n, a(n) for n=2..2000
• Cyril Banderier, Conjecture checked for n<1000 [It has been reported that this data contains errors]
• Pierre CAMI, PFGW Script
• Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.

Crossrefs

Cf. A002110, A005235.

Programs

Maple

for n from 2 to 60 do printf(`%d,`,product(ithprime(j), j=1..n) - prevprime(product(ithprime(j), j=1..n)-1)) od:

Mathematica

PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; k ]; Primorial[ n_Integer ] := Module[ {k = Product[ Prime[ j ], {j, 1, n} ]}, k ]; LF[ n_Integer ] := (p = Primorial[ n ] - 1; q = PrevPrime[ p ]; p - q + 1); Table[ LF[ n ], {n, 2, 60} ]
a[2]=3; a[n_] := (For[m=(Prime[n+1]+1)/2, !PrimeQ[Product[Prime[k], {k, n}] - 2m+1], m++ ]; 2m-1); Table[a[n], {n, 2, 60}]

Formula

a(n) = 1 + the difference between the n-th primorial less one and the previous prime.

From Pierre CAMI, Aug 19 2017: (Start)

Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} prime(n)) = Pi/2.

Floor(a(n) / prime(n)) is always < 8. (End)

Conjecture: Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} prime(n)) = 3/2. - Alain Rocchelli, Nov 07 2022