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

2,1

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

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.

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

Example

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

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}]

Crossrefs

Cf. A002110, A005235.

Sequence in context: A020574 A020618 A184865 * A183176 A045417 A260379

Adjacent sequences: A055208 A055209 A055210 * A055212 A055213 A055214

Keyword

nonn

Author

Robert G. Wilson v, Jul 04 2000

Status

approved