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A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

%I A035346 #54 Nov 09 2024 09:56:29
%S A035346 1,2,3,6,7,8,14,16,17,21,73,801,1971,3332,3469,3509,4318,7986,41292
%N A035346 Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).
%C A035346 Positive n such that A002110(n) + A000040(n+1) is prime. - _Robert Israel_, Dec 02 2015
%C A035346 Subsequence of A265109. - _Altug Alkan_, Dec 02 2015
%H A035346 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 A035346 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 A035346 a(10) = 21 because A002110(21) + prime(22) = 40729680599249024150621323549 = 2*3*5*...*67*71*73 + 79 is prime.
%p A035346 p:= 3:
%p A035346 A[1]:= 1:
%p A035346 count:= 1:
%p A035346 Primorial:= 2:
%p A035346 for n from 2 to 1000 do
%p A035346   Primorial:= Primorial*p;
%p A035346   p:= nextprime(p);
%p A035346   if isprime(Primorial + p) then
%p A035346     count:= count+1;
%p A035346     A[count]:= n;
%p A035346   fi
%p A035346 od:
%p A035346 seq(A[i],i=1..count); # _Robert Israel_, Dec 02 2015
%t A035346 Select[Range@ 801, PrimeQ[Product[Prime@ k, {k, #}] + Prime[# + 1]] &] (* _Michael De Vlieger_, Dec 02 2015 *)
%o A035346 (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 A035346 Cf. A000040, A002110, A005235, A006862, A035345, A265109.
%K A035346 nonn,more
%O A035346 1,2
%A A035346 _N. J. A. Sloane_
%E A035346 a(10)-a(11) were found by _Labos Elemer_, May 02 2000
%E A035346 a(12) from _Ralf Stephan_, Oct 20 2002
%E A035346 Offset changed by _Altug Alkan_, Dec 02 2015
%E A035346 a(13) from _Michael De Vlieger_, Dec 02 2015
%E A035346 a(14)-a(18) from _Altug Alkan_, Dec 02 2015
%E A035346 a(19) from _Henri Lifchitz_, Nov 08 2024