A276192 - OEIS

%I #18 Aug 29 2016 19:13:37

%S 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 123,131,134,149,161,172,175,221,229,345,353,613

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

%C 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 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 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 a(37) > 10^8. - _Dana Jacobsen_, Aug 29 2016

%e 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 t[n_] := n(n+1)/2;

%t is[n_] := !Or@@Table[PrimeQ[k] && PrimeQ[k+2], {k, t[n], t[n+1]-3}];

%t Select[Range[700], is] (* _Andrey Zabolotskiy_, Aug 24 2016 *)

%o (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 (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 Cf. A000217, A001097, A091592.

%K nonn,more

%O 1,2

%A _Altug Alkan_, Aug 24 2016