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Primes such that the ratio between the distance to the next prime and from the previous prime appears for the first time.
5

%I #39 Feb 26 2018 18:58:49

%S 3,5,11,23,29,31,37,89,113,127,139,149,199,251,293,331,337,367,409,

%T 521,523,631,701,787,797,953,1087,1129,1151,1259,1277,1327,1361,1381,

%U 1399,1657,1669,1847,1933,1949,1951,1973,2477,2503,2579,2633,2861,2879,2971,2999,3089,3137,3163,3229,3407

%N Primes such that the ratio between the distance to the next prime and from the previous prime appears for the first time.

%C Number of terms less than 10^n: 2, 8, 26, 85, 224, 511, 1035, 1905, 3338, ..., . - _Robert G. Wilson v_, Nov 30 2016

%H Robert G. Wilson v, <a href="/A275785/b275785.txt">Table of n, a(n) for n = 1..7589</a>

%e a(1) = 3 because this is the first prime for which it is possible to determine the ratio between the distance to the next prime (5) and from the previous prime (2). This first ratio is 2.

%e a(2) = 5 because the ratio between the distance to the next prime (7) and from the previous prime (3) is 1 and this ratio has not appeared before.

%e The third element a(3) is not 7 because (11-7)/(7-5) = 2, a ratio that appeared before with a(1), so a(3) = 11 because (13-11)/(11-7) = 1/2, a ratio that did not appear before.

%t nmax = 720;

%t a = Prime[Range[nmax]];

%t gaps = Rest[a] - Most[a];

%t gapsratio = Rest[gaps]/Most[gaps];

%t newpindex = {}; newgratios = {}; i = 1;

%t While[i < Length[gapsratio] + 1,

%t If[Cases[newgratios, gapsratio[[i]]] == {},

%t AppendTo[newpindex, i + 1];

%t AppendTo[newgratios, gapsratio[[i]]] ];

%t i++];

%t Prime[newpindex]

%t p = 2; q = 3; r = 5; rtlst = qlst = {}; While[q < 10000, rt = (r - q)/(q - p); If[ !MemberQ[rtlst, rt], AppendTo[rtlst, rt]; AppendTo[qlst, q]]; p = q; q = r; r = NextPrime@ r]; qlst (* _Robert G. Wilson v_, Nov 30 2016 *)

%Y Cf. A168253, A179210, A179234, A179256, A274263, A276309, A276812.

%K nonn

%O 1,1

%A _Andres Cicuttin_, Nov 14 2016