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A339314 a(n) is the least semiprime k > n-th semiprime s = A001358(n) such that k-s and k+s are both semiprimes.

%I A339314 #36 Mar 12 2022 14:09:14
%S A339314 10,15,86,25,35,106,25,55,94,51,58,85,39,77,94,95,74,55,106,178,143,
%T A339314 155,69,118,95,142,121,118,119,91,146,142,115,206,115,115,206,169,134,
%U A339314 146,143,178,133,158,155,262,177,158,178,155,159,183,254,194,205,202,226,187,298,206,226,209
%N A339314 a(n) is the least semiprime k > n-th semiprime s = A001358(n) such that k-s and k+s are both semiprimes.
%C A339314 Differences k - s: 6, 9, 77, 15, 21, 91, 4, 33, 69, 25, ... with minimal value 4.
%C A339314 What about the maximal value of k - s?
%C A339314 k-s is unbounded, because the gaps between semiprimes are unbounded. In fact, given any n distinct primes, by the Chinese Remainder Theorem there exist n consecutive positive integers that are each divisible by the cube of one of these primes (and thus not semiprimes). - _Robert Israel_, Dec 27 2020
%H A339314 Robert Israel, <a href="/A339314/b339314.txt">Table of n, a(n) for n = 1..10000</a>
%e A339314 s=4, k=10, 6 and 14 are all semiprimes,
%e A339314 s=6, k=15, 9 and 21 are all semiprimes,
%e A339314 s=9, k=86, 77 and 95 are all semiprimes.
%p A339314 N:= 10^3:
%p A339314 SP:= select(t -> numtheory:-bigomega(t)=2, [$4..N]):
%p A339314 f:= proc(n) local i,s;
%p A339314   s:= SP[n];
%p A339314   for i from n+1 do
%p A339314     if numtheory:-bigomega(SP[i]-s)=2 and numtheory:-bigomega(SP[i]+s)=2 then return SP[i] fi
%p A339314   od;
%p A339314 end proc:
%p A339314 map(f, [$1..100]); # _Robert Israel_, Dec 27 2020
%o A339314 (PARI) issemip(n) = bigomega(n)==2;
%o A339314 lista(nn) = {my(v = select(issemip, [1..nn])); for (n=1, #v, my(ik=n+1, s=v[n]); while (!(issemip(v[ik]+s) && issemip(v[ik]-s)), ik++; if (ik>#v, return)); print1(v[ik], ", "););} \\ _Michel Marcus_, Dec 19 2020
%Y A339314 Cf. A001358.
%K A339314 nonn
%O A339314 1,1
%A A339314 _Zak Seidov_, Dec 17 2020