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

1,1

Comments

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

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A262431 A262085 A015915 * A106901 A154550 A023202

Adjacent sequences: A275782 A275783 A275784 * A275786 A275787 A275788

Keyword

nonn

Author

Andres Cicuttin, Nov 14 2016

Status

approved