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

Author

Vladimir Shevelev, Apr 25 2014

Keywords

• nonn

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

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Wikipedia, Chen's theorem

Crossrefs

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

Programs

Mathematica

sk[s_]:=Module[{k=1},While[!PrimeQ[s+k]||!PrimeQ[s-k],k++];k]; sk/@Select[Range[300],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 07 2025 *)

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

Extensions

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