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A276192 Numbers n such that there is no twin prime pair between A000217(n) and A000217(n+1) (n > 0).

%I A276192 #18 Aug 29 2016 19:13:37
%S A276192 1,3,6,9,12,15,17,26,27,30,32,36,37,38,42,43,48,51,55,65,69,75,77,108,
%T A276192 123,131,134,149,161,172,175,221,229,345,353,613
%N A276192 Numbers n such that there is no twin prime pair between A000217(n) and A000217(n+1) (n > 0).
%C A276192 Numbers n such that there is no pair of twin primes p, p+2 with n*(n+1)/2 <= p < p+2 < (n+1)*(n+2)/2.
%C A276192 Number of twin prime pairs between A000217(n) and A000217(n+1) are 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 3, 1, 0, 0, 2, 2, 1, 1, 0, 1, 4, 0, 1, 2, 1, 0, 2, 2, 1, 1, 2, 2, 1, 1, 5, 0, 2, 2, 1, 0, 1, 1, 2, 2, 2, 0, 2, 0, 1, 1, 3, 4, 2, 3, ...
%C A276192 Probably the sequence is finite, and a(36)=613 is the last term. If a(37) exists, then a(37)>10000. - _Andrey Zabolotskiy_, Aug 24 2016
%C A276192 a(37) > 10^8. - _Dana Jacobsen_, Aug 29 2016
%e A276192 3 is a term because there is no twin prime pair between A000217(3) = 6 and A000217(4) = 10, even though 7 is one of a prime pair and between 6 and 10, 5 isn't so the pair doesn't exclude 3.
%t A276192 t[n_] := n(n+1)/2;
%t A276192 is[n_] := !Or@@Table[PrimeQ[k] && PrimeQ[k+2], {k, t[n], t[n+1]-3}];
%t A276192 Select[Range[700], is] (* _Andrey Zabolotskiy_, Aug 24 2016 *)
%o A276192 (Perl) use ntheory ":all"; sub is_a276192 { my $n=shift; my $t=($n*$n+$n)>>1; twin_prime_count($t,$t+$n+1-2) == 0; } # _Dana Jacobsen_, Aug 29 2016
%o A276192 (Perl) use ntheory ":all"; sub is_a276192 { my($n,$t,$e,$p,$prev)=(shift); $t = ($n*$n+$n)>>1; $e=$t+$n+1-2; $p = next_prime($t-1); $prev = next_prime($p); ($prev, $p) = ($p, next_prime($p)) while ($p-$prev) != 2;  $prev > $e; }    my $n=1; for (1..36) { $n++ until is_a276192($n); say "$_ ",$n++; } # _Dana Jacobsen_, Aug 29 2016
%Y A276192 Cf. A000217, A001097, A091592.
%K A276192 nonn,more
%O A276192 1,2
%A A276192 _Altug Alkan_, Aug 24 2016