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A283371 Maximum number of pairs of primes (p, q) such that p < q = < prime (n) and q - p = constant.

%I A283371 #9 Mar 06 2017 19:43:34
%S A283371 0,1,1,2,2,3,3,4,5,6,6,7,7,8,9,10,11,11,12,12,13,14,14,15,15,15,16,17,
%T A283371 18,19,19,19,20,21,21,21,22,22,23,23,24,25,25,26,27,27,28,29,30,31,31,
%U A283371 31,32,32,33,34,35,36,36,37,37,38,39,40,41,41,41,42,43,43,43,43,44,44,45,46,47,48,48,49,50,50
%N A283371 Maximum number of pairs of primes (p, q) such that p < q = < prime (n) and q - p = constant.
%C A283371 Maximum number of different ways of expressing a positive number as a difference of two distinct primes less than or equal to prime(n).
%C A283371 Is there any n such that a(n+1) - a(n) > 1?
%C A283371 What is the asymptotic behavior of a(n)?
%C A283371 To answer the first question: for all n, either a(n+1) = a(n) or a(n+1) = a(n) + 1. - _Charles R Greathouse IV_, Mar 06 2017
%H A283371 Charles R Greathouse IV, <a href="/A283371/b283371.txt">Table of n, a(n) for n = 1..10000</a>
%F A283371 a(n) >> n/log n. In particular, lim inf a(n) * (log n)/n >= 1/2. - _Charles R Greathouse IV_, Mar 06 2017
%e A283371 a(1)=0 because there are no two distinct primes less than or equal to prime(1)=2.
%e A283371 a(2)=1 because there are only two distinct primes less than or equal to prime(2)=3, and then there is only one positive difference among them: 3 - 2 = 1.
%e A283371 a(3)=1 because the three pairs of distinct primes less than or equal to prime(3)=5, i.e., (2,3), (3,5), and (2,5), produce different positive differences: 3 - 2 = 1, 5 - 3 = 2, and 5 - 2 = 3.
%e A283371 a(4)=2 because among all pairs of distinct primes taken from the first four primes, 2, 3, 5, and 7, there are two pairs with same positive difference, i.e., 7 - 5 = 5 - 3 = 2.
%e A283371 a(6)=3 because among all pairs of distinct primes taken from the first six primes, 2, 3, 5, 7, 11, and 13, there are at most three pairs with the same positive difference, i.e., 13 - 11 = 7 - 5 = 5 - 3 = 2.
%t A283371 a[n_]:=Module[{fp,fps,fpst,fpstts,fpsttst},
%t A283371 fp=Prime[Range[n]];
%t A283371 fps=Subsets[fp,{2}];
%t A283371 fpst=Table[Abs@(fps[[j]][[2]]-fps[[j]][[1]]),{j,1,Length[fps]}];
%t A283371 fpstts=fpst//Tally;
%t A283371 If[n<2,0,fpsttst=fpstts//Transpose;fpsttst[[2]]//Max]//Return];
%t A283371 Table[a[n],{n,1,120}]
%o A283371 (PARI) first(n)=my(v=vector(n),P=primes(n),H=vectorsmall((P[#P]-P[2])/2,i,0)); v[2]=1; for(n=3,#P, for(i=2,n-1,H[(P[n]-P[i])/2]++); v[n]=vecmax(H)); v \\ _Charles R Greathouse IV_, Mar 06 2017
%Y A283371 Cf. A283302, A030173.
%K A283371 nonn
%O A283371 1,4
%A A283371 _Andres Cicuttin_, Mar 06 2017