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

1, 3, 6, 9, 12, 15, 17, 26, 27, 30, 32, 36, 37, 38, 42, 43, 48, 51, 55, 65, 69, 75, 77, 108, 123, 131, 134, 149, 161, 172, 175, 221, 229, 345, 353, 613

Offset

1,2

Comments

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.

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, ...

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

a(37) > 10^8. - Dana Jacobsen, Aug 29 2016

Links

Table of n, a(n) for n=1..36.

Example

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.

Mathematica

t[n_] := n(n+1)/2;
is[n_] := !Or@@Table[PrimeQ[k] && PrimeQ[k+2], {k, t[n], t[n+1]-3}];
Select[Range[700], is] (* Andrey Zabolotskiy, Aug 24 2016 *)

Prog

(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
(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

Crossrefs

Cf. A000217, A001097, A091592.

Sequence in context: A162500 A191405 A198263 * A347404 A329771 A282143

Adjacent sequences: A276189 A276190 A276191 * A276193 A276194 A276195

Keyword

nonn,more

Author

Altug Alkan, Aug 24 2016

Status

approved