A360406 - OEIS

%I #17 Feb 22 2023 08:08:29

%S 1,1,9,14,31,826,1,34

%N 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.

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

%C a(11) = 692, a(12) = 8, a(13) = 792. - _Robert Israel_, Feb 22 2023

%e a(1) = 1 as prime(1) * prime(2) - 1 = 2 * 3 - 1 = 5, which is divisible by prime(3) = 5.

%e a(2) = 1 as prime(2) * prime(3) - 1 = 3 * 5 - 1 = 14, which is divisible by prime(4) = 7.

%e a(3) = 9 as prime(3) * ... * prime(12) - 1 = 1236789689134, which is divisible by prime(13) = 41.

%p f:= proc(n) local P,k,p;

%p P:= ithprime(n); p:= nextprime(P);

%p for k from 0 to 10^6 do

%p   if P-1 mod p = 0 then return k fi;

%p   p:= nextprime(p);

%p  od;

%p FAIL

%p end proc:

%p map(f, [$1..8]); # _Robert Israel_, Feb 22 2023

%o (Python)

%o from sympy import prime, nextprime

%o def A360406(n):

%o     p = prime(n)

%o     q = nextprime(p)

%o     s, k = p*q, 1

%o     while (s-1)%(q:=nextprime(q)):

%o         k += 1

%o         s *= q

%o     return k # _Chai Wah Wu_, Feb 06 2023

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

%K nonn,more

%O 1,3

%A _Scott R. Shannon_, Feb 06 2023