succesive natural numbers as records

[Artur]

Dear Seqfans,
Is little strange for me that records in A127267 are successive natural  
numbers up to 100000

%I A127267
%S A127267 0, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2,  
2, 2, 2, 1, 1,
1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2,  
1,
2, 2, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3,  
2,
3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2
%N A127267 Number of primes between x and x - x^(23/42)
%C A127267 This sequence visualised proof Hardy and Wright from 1979 p.  
415, that for x>1 existed one or more prime between x and x - x^(23/42)
%D A127267 Hardy, G. H. and Wright, W. M. "Unsolved Problems Concerning  
Primes." §2.8 and Appendix §3 in An Introduction to the Theory of  
Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 19 and  
415-416, 1979.
%H A127267 <a  
href="http://mathworld.wolfram.com/LegendresConjecture.html">More  
information</a>
%t A127267 Table[PrimePi[x] - PrimePi[N[x - x^(23/42)]], {x, 1, 100}]
%Y A127267 Cf.  A127079, A127259, A127260, A127262, A127263, A127264
%O A127267 1
%K A127267 ,nonn,
%A A127267 Artur Jasinski (grafix at csl.pl), Jan 10 2007


%I A127268
%S A127268 2, 3, 43, 107, 109, 241, 467, 619, 1051, 1307, 1321, 1327,  
2399, 2713, 2719,
2731, 5009, 5023, 6367, 7577, 7583, 7607, 11941, 12547, 14783, 14891,  
17491,
17497, 19571, 19577, 19583, 19597, 26893, 27961, 28661, 28663, 32587,  
42223,
42227, 42467, 42473, 42499, 51719, 51721, 57287, 59281, 63667, 63691,  
63697,
63703, 63737, 77711, 77713, 81083, 88019, 92821, 92863, 92867
%N A127268 Values x where records occured in A127267 (up to 100000 these  
records are succesive natura numbers A000027)
%C A127268 Sequence after proof Hardy and Wright from 1979 p. 415, that  
for x>1 existed one or more prime between x and x - x^(23/42)
%t A127268 max = 0; b = {}; a = {};
    Do[If[PrimePi[x] - PrimePi[N[x - x^(
     23/42)]] > max, max =
         PrimePi[x] - PrimePi[N[x - x^(23/42)]]; AppendTo[a, PrimePi[
       x] - PrimePi[N[x - x^(
       23/42)]]]; AppendTo[b, x]], {x, 1, 100000}]; Print[a]; Print[b]
%Y A127268 Cf.  A127079, A127259, A127260, A127262, A127263, A127264,  
A127265, A127265, A127267
%O A127268 1
%K A127268 ,nonn,
%A A127268 Artur Jasinski (grafix at csl.pl), Jan 10 2007