A290585 a(n) is the largest number <= n such that 1 + a(1)*a(2)*...*a(n) is prime.

1, 2, 3, 3, 4, 6, 6, 4, 7, 7, 3, 10, 13, 12, 10, 9, 13, 14, 15, 16, 13, 21, 22, 11, 25, 26, 27, 17, 29, 23, 7, 11, 30, 24, 34, 1, 1, 1, 1, 1, 1, 1, 1, 1, 45, 39, 23, 48, 32, 25, 44, 49, 53, 31, 1, 1, 1, 1, 59, 46, 53, 55, 62, 40, 62, 59, 46, 41, 9, 62, 59, 64, 1, 1, 1, 1, 1, 1, 1, 80, 57, 78, 80, 1, 85

Offset: 1

Author

Thomas Ordowski, Aug 07 2017

Keywords

• nonn

Comments

a(n) = n for n = 1, 2, 3, 6, 13, 25, 26, 27, 29, 45, 48, 53, 59, 80, 85, ...

If a(n) = 1, then the next entry > 1 is a(m) = m for the least m > n such that 1 + m * Product_{j=1..n-1} a_j is prime. By Dirichlet's theorem such m exists. - Robert Israel, Aug 07 2017

Links

• Iain Fox, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A036012, A290639.

Programs

Maple

A[1]:= 1: P:= 1:
for n from 2 to 200 do
  for k from n to 0 by -1 do
    if isprime(1+k*P) then
      A[n]:= k;
      P:= P*k;
      break
    fi
  od;
od:
seq(A[i],i=1..200); # Robert Israel, Aug 07 2017

Mathematica

p = 1; Table[t = SelectFirst[Range[n, 1, -1], PrimeQ[1 + p #] &]; p *= t; t, {n, 85}] (* Giovanni Resta, Aug 08 2017 *)

PARI

first(n) = { my(i = 1, res = vector(n)); res[1]=1; for(x=2, n, forstep(k=x, 0, -1, if(ispseudoprime(1+k*i), res[x]=k; i*=k; break()))); res; } \\ Iain Fox, Nov 15 2017

Python

from sympy import isprime
A=[0, 1]
p=1
for n in range(2, 201):
    for k in range(n, -1, -1):
        if isprime(1 + k*p):
            A.append(k)
            p*=k
            break
print(A[1:]) # Indranil Ghosh, Aug 10 2017

Extensions

More terms from Robert Israel, Aug 07 2017