A239020 - OEIS

%I #25 Nov 07 2024 21:55:20

%S 4,3,2,15,6,2,150,75,20,6,78,85,2490,30,18,195,5160,490,330,12,2,870,

%T 330,13,42,105,2280,375,12,41,1632,720,90,3,216,2,1380,615,98,84,438,

%U 65,600,210,148,735,3870,115,138,39,182,2715,16590,48,60,63,210,120

%N Smallest number k such that k*n +/- 1 and k*n^2 +/- 1 are two sets of twin primes. a(n) = 0 if no such number exists.

%C If n>3 is odd and not a multiple of 3, then a(n) is a multiple of 6; e.g., a(5) = 6, a(7) = 150, a(11) = 78. If n>3 is even and not a multiple of 3, then a(n) is a multiple of 3. In short, for n>1, k*n should be a multiple of 6. - _Zak Seidov_, Mar 13 2014

%e 1*2 +/- 1 (1 and 3) and 1*2^2 +/- 1 (3 and 5) are not two sets of twin primes. 2*2 +/- 1 (3 and 5) and 2*2^2 +/- 1 (7 and 9) are not two sets of twin primes. However, 3*2 +/- 1 (5 and 7) and 3*2^2 +/- 1 (11 and 13) are two sets of twin primes. Thus, a(2) = 3.

%o (Python)

%o from sympy import isprime

%o def b(n):

%o   for k in range(10**5):

%o     if isprime(k*n+1) and isprime(k*n-1) and isprime(k*(n**2)+1) and isprime(k*(n**2)-1):

%o       return k

%o n = 1

%o while n < 100:

%o   print(b(n))

%o   n += 1

%o (PARI) a(n) = {k = 1; while (! (isprime(k*n+1) && isprime(k*n-1) && isprime(k*n^2+1) && isprime(k*n^2-1)), k++); k;} \\ _Michel Marcus_, Mar 15 2014

%Y Cf. A231819, A053989, A035092, A034693.

%K nonn

%O 1,1

%A _Derek Orr_, Mar 09 2014