A275785 Primes such that the ratio between the distance to the next prime and from the previous prime appears for the first time.

3, 5, 11, 23, 29, 31, 37, 89, 113, 127, 139, 149, 199, 251, 293, 331, 337, 367, 409, 521, 523, 631, 701, 787, 797, 953, 1087, 1129, 1151, 1259, 1277, 1327, 1361, 1381, 1399, 1657, 1669, 1847, 1933, 1949, 1951, 1973, 2477, 2503, 2579, 2633, 2861, 2879, 2971, 2999, 3089, 3137, 3163, 3229, 3407

Offset: 1

Author

Andres Cicuttin, Nov 14 2016

Keywords

• nonn

Comments

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

Examples

			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.
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.
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.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..7589

Crossrefs

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

Programs

Mathematica

nmax = 720;
a = Prime[Range[nmax]];
gaps = Rest[a] - Most[a];
gapsratio = Rest[gaps]/Most[gaps];
newpindex = {}; newgratios = {}; i = 1;
While[i < Length[gapsratio] + 1,
If[Cases[newgratios, gapsratio[[i]]] == {},
  AppendTo[newpindex, i + 1];
  AppendTo[newgratios, gapsratio[[i]]] ];
  i++];
Prime[newpindex]
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 *)