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%I A055211 #40 Dec 15 2022 14:15:09
%S A055211 3,7,11,13,17,29,23,43,41,73,59,47,89,67,73,107,89,101,127,97,83,89,
%T A055211 97,251,131,113,151,263,251,223,179,389,281,151,197,173,239,233,191,
%U A055211 223,223,293,593,293,457,227,311,373,257,307,313,607,347,317,307,677,467
%N A055211 Lesser Fortunate numbers.
%C A055211 a(1) is not defined. The first 1000 terms are all prime and it is conjectured that all terms are primes.
%C A055211 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
%H A055211 Pierre CAMI, <a href="/A055211/b055211.txt">Table of n, a(n) for n=2..2000</a>
%H A055211 Cyril Banderier, <a href="http://algo.inria.fr/banderier/Computations/prime_factorial.html">Conjecture checked for n<1000</a> [It has been reported that this data contains errors]
%H A055211 Pierre CAMI, <a href="/A055211/a055211.txt">PFGW Script</a>
%H A055211 Antonín Čejchan, Michal Křížek, and Lawrence Somer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Krizek/krizek3.html">On Remarkable Properties of Primes Near Factorials and Primorials</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
%F A055211 a(n) = 1 + the difference between the n-th primorial less one and the previous prime.
%F A055211 From _Pierre CAMI_, Aug 19 2017: (Start)
%F A055211 Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} prime(n)) = Pi/2.
%F A055211 Floor(a(n) / prime(n)) is always < 8. (End)
%F A055211 Conjecture: Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} prime(n)) = 3/2. - _Alain Rocchelli_, Nov 07 2022
%e A055211 a(3) = 7 since 2*3*5 = 30, 30-1 = 29, previous prime is 23, 30-23 = 7.
%p A055211 for n from 2 to 60 do printf(`%d,`,product(ithprime(j), j=1..n) - prevprime(product(ithprime(j), j=1..n)-1)) od:
%t A055211 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} ]
%t A055211 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}]
%Y A055211 Cf. A002110, A005235.
%K A055211 nonn
%O A055211 2,1
%A A055211 _Robert G. Wilson v_, Jul 04 2000

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