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Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).
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%I #48 May 22 2022 13:59:42

%S 1,2,3,6,7,8,14,16,17,21,73,801,1971,3332,3469,3509,4318,7986

%N Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

%C Positive n such that A002110(n) + A000040(n+1) is prime. - _Robert Israel_, Dec 02 2015

%C Subsequence of A265109. - _Altug Alkan_, Dec 02 2015

%H 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.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2689634">The evidence for Fortune's conjecture</a>, Math. Mag. 54 (1981), 209-210.

%e a(10) = 21 because A002110(21) + prime(22) = 40729680599249024150621323549 = 2*3*5*...*67*71*73 + 79 is prime.

%p p:= 3:

%p A[1]:= 1:

%p count:= 1:

%p Primorial:= 2:

%p for n from 2 to 1000 do

%p Primorial:= Primorial*p;

%p p:= nextprime(p);

%p if isprime(Primorial + p) then

%p count:= count+1;

%p A[count]:= n;

%p fi

%p od:

%p seq(A[i],i=1..count); # _Robert Israel_, Dec 02 2015

%t Select[Range@ 801, PrimeQ[Product[Prime@ k, {k, #}] + Prime[# + 1]] &] (* _Michael De Vlieger_, Dec 02 2015 *)

%o (PARI) lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(k+1)), print1(k, ", ")); ); } \\ _Altug Alkan_, Dec 02 2015

%Y Cf. A000040, A002110, A005235, A006862, A035345, A265109.

%K nonn,more

%O 1,2

%A _N. J. A. Sloane_

%E The terms 21 and 73 were found by _Labos Elemer_, May 02 2000

%E One more term from _Ralf Stephan_, Oct 20 2002

%E Offset changed by _Altug Alkan_, Dec 02 2015

%E Term 1971 from _Michael De Vlieger_, Dec 02 2015

%E Terms 3332, 3469, 3509, 4318, 7986 from _Altug Alkan_, Dec 02 2015