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

2, 3, 5, 7, 11, 11, 16, 15, 21, 22, 30, 36, 41, 43, 34, 36, 56, 60, 48, 55, 54, 59, 57, 75, 42, 93, 93, 103, 104, 75, 126, 123, 133, 129, 148, 104, 146, 162, 159, 128, 177, 159, 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

Offset: 1

Author

Thomas Ordowski, Aug 08 2017

Keywords

• nonn

Comments

a(n) = prime(n) for n = 1, 2, 3, 4, 5, 13, 14, ...

If a(n) = 1 and a(n+1) > 1, then prime(n) < a(n+1) <= prime(n+1).

Conjecture: a(n) > 1 for every n. - Thomas Ordowski, Aug 08 2017

Indeed, a(n) > n for all n <= 460. - Robert Israel, Aug 08 2017

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A036012, A290585.

Programs

Maple

A[1]:= 2: P:= 2:
for n from 2 to 200 do
  for k from ithprime(n) by -1 do
    if isprime(1+P*k) then A[n]:= k; P:= P*k; break fi
  od;
od:
seq(A[i],i=1..200); # Robert Israel, Aug 08 2017

Mathematica

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 *)
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 *)

Extensions

More terms from Robert Israel, Aug 08 2017