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a(n) is the smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is prime.
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%I #43 Feb 13 2024 06:56:10

%S 1,2,3,5,6,9,12,16,22,25,29,31,35,47,57,61,66,79,81,108,114,148,163,

%T 172,185,198,203,205,236,265,275,282,294,312,344,359,377,397,398,411,

%U 427,431,493,512,589,647,648,660,708,719,765,887,911,916,935,1062,1093

%N a(n) is the smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is prime.

%D H. Dubner, Recursive Prime Generating Sequences, Table 4 pp. 173 Journal of Recreational Mathematics 29(3) 1998 Baywood NY.

%H Charles R Greathouse IV and T. D. Noe, <a href="/A046966/b046966.txt">Table of n, a(n) for n = 1..500</a> (first 200 terms from Noe)

%e 1*2*3*5 + 1 = 31 is prime.

%t a[1] = 1; p[1] = 1;

%t a[n_] := a[n] = For[an = a[n-1] + 1, True, an++, pn = p[n-1]*an; If[ PrimeQ[pn+1], p[n] = pn; Return[an] ] ];

%t Table[a[n], {n, 1, 60}]

%t (* _Jean-François Alcover_, Sep 17 2012 *)

%t Module[{cc={1},k},Do[k=Last[cc]+1;While[!PrimeQ[Times@@Join[cc,{k}]+1], k++];AppendTo[cc,k],{60}];cc] (* _Harvey P. Dale_, Jan 21 2013 *)

%t nxt[{t_,a_}]:=Module[{k=a+1},While[CompositeQ[t*k+1],k++];{t*k,k}]; NestList[nxt,{1,1},60][[All,2]] (* _Harvey P. Dale_, May 22 2021 *)

%o (PARI) first(n)=my(v=vector(n),N=1,t=1); v[1]=1; for(k=2,n, while(!ispseudoprime(1 + N*t++),); N*=v[k]=t); v \\ _Charles R Greathouse IV_, Apr 07 2020

%Y Cf. A046972.

%K nonn,nice

%O 1,2

%A _G. L. Honaker, Jr._

%E More terms from _Jason Earls_, Jan 25 2002

%E Definition corrected by _T. D. Noe_, Feb 14 2007