A360406 a(n) = minimal positive k such that prime(n) * prime(n+1) * ... * prime(n+k) - 1 is divisible by prime(n+k+1), or -1 if no such k exists.

1, 1, 9, 14, 31, 826, 1, 34

Offset: 1

Author

Scott R. Shannon, Feb 06 2023

Keywords

• nonn
• more

Comments

Assuming a(9) exists it is greater than 1.75 million.

a(11) = 692, a(12) = 8, a(13) = 792. - Robert Israel, Feb 22 2023

Examples

			a(1) = 1 as prime(1) * prime(2) - 1 = 2 * 3 - 1 = 5, which is divisible by prime(3) = 5.
a(2) = 1 as prime(2) * prime(3) - 1 = 3 * 5 - 1 = 14, which is divisible by prime(4) = 7.
a(3) = 9 as prime(3) * ... * prime(12) - 1 = 1236789689134, which is divisible by prime(13) = 41.
		

Crossrefs

Cf. A360376, A360297, A000040, A007504, A332542, A332580.

Programs

Maple

f:= proc(n) local P,k,p;
P:= ithprime(n); p:= nextprime(P);
for k from 0 to 10^6 do
  if P-1 mod p = 0 then return k fi;
  p:= nextprime(p);
 od;
FAIL
end proc:
map(f, [$1..8]); # Robert Israel, Feb 22 2023

Python

from sympy import prime, nextprime
def A360406(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p*q, 1
    while (s-1)%(q:=nextprime(q)):
        k += 1
        s *= q
    return k # Chai Wah Wu, Feb 06 2023