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A290639 a(n) = largest number <= prime(n) such that 1 + a(1)*a(2)*...*a(n) is prime.

%I A290639 #27 Jul 27 2025 14:50:40
%S A290639 2,3,5,7,11,11,16,15,21,22,30,36,41,43,34,36,56,60,48,55,54,59,57,75,
%T A290639 42,93,93,103,104,75,126,123,133,129,148,104,146,162,159,128,177,159,
%U A290639 153,175,184,187,193,223,210,151,164,170,240,239,254,261,201,261,253,254,170,255,297,257,270,291,309,267,341,310,261,316,363,329,373,361,327,381,373,401,346,351,379
%N A290639 a(n) = largest number <= prime(n) such that 1 + a(1)*a(2)*...*a(n) is prime.
%C A290639 a(n) = prime(n) for n = 1, 2, 3, 4, 5, 13, 14, ...
%C A290639 If a(n) = 1 and a(n+1) > 1, then prime(n) < a(n+1) <= prime(n+1).
%C A290639 Conjecture: a(n) > 1 for every n. - _Thomas Ordowski_, Aug 08 2017
%C A290639 Indeed, a(n) > n for all n <= 460. - _Robert Israel_, Aug 08 2017
%H A290639 Amiram Eldar, <a href="/A290639/b290639.txt">Table of n, a(n) for n = 1..1000</a>
%p A290639 A[1]:= 2: P:= 2:
%p A290639 for n from 2 to 200 do
%p A290639   for k from ithprime(n) by -1 do
%p A290639     if isprime(1+P*k) then A[n]:= k; P:= P*k; break fi
%p A290639   od;
%p A290639 od:
%p A290639 seq(A[i],i=1..200); # _Robert Israel_, Aug 08 2017
%t A290639 a[1] = 2; a[n_] := a[n] = Module[{k = Prime[n], r = Product[a[i], {i, 1, n - 1}]}, While[! PrimeQ[1 + k*r], k--]; k]; Array[a, 100] (* _Amiram Eldar_, Jan 19 2023 *)
%t A290639 nxt[{n_,p_,a_}]:=Module[{k=Prime[n+1]},While[!PrimeQ[1+p*k],k--];{n+1,p*k,k}]; NestList[nxt,{1,2,2},85][[;;,3]] (* _Harvey P. Dale_, Jul 27 2025 *)
%Y A290639 Cf. A000040, A036012, A290585.
%K A290639 nonn
%O A290639 1,1
%A A290639 _Thomas Ordowski_, Aug 08 2017
%E A290639 More terms from _Robert Israel_, Aug 08 2017