A241539 Smallest k>=1 such that the n-th semiprime + or - k are both primes, or a(n)=0 if there is no such k.

1, 1, 2, 3, 3, 2, 2, 9, 6, 3, 4, 3, 6, 9, 2, 15, 12, 8, 12, 4, 15, 9, 6, 2, 15, 6, 15, 12, 3, 14, 12, 4, 15, 6, 3, 2, 12, 9, 12, 18, 9, 14, 2, 6, 3, 10, 15, 6, 6, 33, 18, 9, 8, 12, 15, 12, 4, 15, 10, 6, 6, 3, 10, 9, 24, 6, 27, 18, 14, 15, 18, 6, 21, 8, 30, 3

Offset

1,3

Comments

If the sequence has no zeros at least for sufficiently large n, then one can believe that, for a pair of sufficiently large semiprimes of the same parity {r,s}, there is a number k=k(r,s) such that either {r-k, s+k} or {r+k, s-k} is a pair of primes. Then, if a representation 2*n = r+s with, say, min{r,s} > log(n) is considered, then, at least for sufficiently large n, it is reduced to the Goldbach representation 2*n = p+q with primes p,q. It is natural to think that a Goldbach-like conjecture that at least every sufficiently large even number is a sum of two semiprimes could be proved more easily than the classic Goldbach conjecture (cf. Chen's theorem).

Links

Table of n, a(n) for n=1..76.

Wikipedia, Chen's theorem

Prog

(PARI)
list(lim) = my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
sp = list(1000); vector(#sp, n, k=1; while(!isprime(sp[n]+k) || !isprime(sp[n]-k), k++); k) \\ Colin Barker, May 31 2014

Crossrefs

Cf. A001358, A241531, A241533, A241535, A241536.

Sequence in context: A030423 A130631 A282014 * A376893 A213512 A218774

Adjacent sequences: A241536 A241537 A241538 * A241540 A241541 A241542

Keyword

nonn

Author

Vladimir Shevelev, Apr 25 2014

Extensions

More terms from Peter J. C. Moses, Apr 28 2014

Status

approved