# Increasing number of twin primes in [k^2,(k+1)^2]

Thu Sep 30 23:39:24 CEST 2004

```SeqFans,

I've submitted a rather speculative sequence somehow illustrating the
numerical evidence for the twin prime conjecture. Counter examples
making the sequence invalid are highly welcome.

%S A099154 122 213 502 545 922 950 749 1098 1330 1450 1634 1623 2135
2110 2177 2244 2760 2413 2556 3280 3454 3211 3740 3540 4104 4096 4391
4457 4592 5309 4758 5720 5747 5295 5902 5456 5920 6395 5810 7007 7109
7450 7540 7170 7586 8420 7658 8630 8095 8409 7884 8661 8560 8906 9016
10347 9107 10595 10036 10454
%N A099154 Largest number k such that the interval [k^2,(k+1)^2]
contains not more than n pairs of twin primes.
%C A099154 The validity of this sequence depends on the twin prime
conjecture.
%H A099154 Eric W. Weisstein's World of Mathematics, <a
href="http://mathworld.wolfram.com/TwinPrimes.html">Twin
Primes.</a>
%H A099154 Eric W. Weisstein's World of Mathematics, <a
href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">Twin
Prime Conjecture.</a>
%e A099154 a(1)=213 because the interval [213^2,214^2]=[45369,45796]
contains one pair of twin primes (45587,45589) wheras all higher
intervals are conjectured to contain at least two pairs of twin primes.
The interval [122^2,123^2]=[A091592(11)^2,(A091592(11)+1)^2] is
conjectured to be the last interval between two consecutive squares
containing no twin primes.
%Y A099154 Cf. A091591 number of pairs of twin primes between n^2 and
(n+1)^2, A091592 numbers n such that there are no twin primes between
n^2 and (n+1)^2, A014574.
%O A099154 0
%K A099154 ,nonn,
%A A099154 Hugo Pfoertner (hugo at pfoertner.org), Sep 30 2004

Hugo

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