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Positive numbers n without a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.
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%I #25 Jun 24 2014 13:18:18

%S 1,16,67,86,131,151,186,191,211,226,541,701

%N Positive numbers n without a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.

%C Conjecture: any integer n > 701 has a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.

%p b:= n-> isprime(6*n-1) and isprime(6*n+1):

%p a:= proc(n) option remember; local i, k, ok;

%p for k from 1 +`if`(n=1, 0, a(n-1)) do ok:= true;

%p for i to iquo(k, 2) while ok

%p do ok:= not(b(i) and b(k-i)) od;

%p if ok then return k fi

%p od

%p end:

%p seq(a(n), n=1..12); # _Alois P. Heinz_, Jun 20 2014

%o (PARI) l=List();a=select(p->isprime(p-2)&&p>5, primes(2000))\6;

%o for(i=1,#a-1,listput(l,2*a[i]);for(j=i+1,#a,listput(l,(a[i]+a[j]))));

%o print(setminus(Set(vector(l[#l]/4, i, i)), Set(l)))

%Y Cf. A002822, A187759.

%K nonn

%O 1,2

%A _Lear Young_, Jun 15 2014