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%I #8 Oct 01 2013 17:58:07
%S 22,11,18,62,59,29,12,47,106,40,17,104,94,44,83,180,22,51,65,60,177,
%T 66,189,51,175,167,112,219,204,198,193,188,36,25,281,328,267,312,305,
%U 249,244,184,175,161
%N First nonzero terms in the sequences formed by the unique count of primes between an and (a+1)n.
%C Care should be taken to choose a sufficiently large n for a given m range for a in the script below. (50,1000) -> 22,11,18,62,59,29,12,47,106,40,17,104,94,44,83,22,.. (50,2000) -> 22,11,18,62,59,29,12,47,106,40,17,104,94,44,83,180,.. Notice the breakdown at the end. While the terms in the sequence tend to oscillate increasing, strange things are possible when more terms are listed. Conjecture: The number of terms in this sequence is infinite.
%F S(a, b) = Sequence of the unique count of primes between an and bn n=1, 2, ...
%e S(1,2) = 22,36,47,79,98,114,134,173.. -> A084141 except for the first term
%e S(2,3) = 11,42,93,110,113,156,186..
%e S(3,4) = 18,100,102,147,200,203,238..
%e 22,11,18 are the first 3 terms in the sequence.
%o (PARI) betanap1n(m,n) = { local(a,c,c1,x,y); v=vector(10002); for(a=1,m, for(x=1,n, c=0; forprime(y=a*x+1,(a+1)*x-1, c++; ); v[x] = c; ); w=vecsort(v); for(x=1,10000, if(w[x]>0, if(w[x+1]<>w[x]&w[x+1]<>w[x+2], print1(w[x]+1",");break); ) ) ) }
%Y Cf. A084141.
%K more,nonn,uned
%O 1,1
%A _Cino Hilliard_, Jan 30 2005
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