A291528 First term s_n(1) of equivalence classes of prime sequences {s_n(k)} for k > 0 derived by records of first differences of Rowland-like recurrences with increasing even starting values e(n) >= 4.

2, 7, 17, 19, 31, 43, 53, 71, 67, 79, 97, 103, 109, 113, 127, 137, 151, 163, 181, 173, 191, 197, 199, 211, 229, 239, 241, 251, 269, 257, 271, 283, 293, 317, 331, 337, 349, 367, 373, 419, 409, 431, 433, 439, 443, 463, 491, 487, 499, 523, 557, 547, 577, 593, 607, 599, 601

Offset

1,1

Comments

These kinds of equivalence classes {s_n(k)} were defined by Shevelev, see Crossrefs.

Some equivalence classes of prime sequences {s_n(k)} have the same tail for a constant C_n < k, such as {s_2(k)} = {a(2),...} = {7,13,29,59,131,...} and {s_5(k)} = {a(5),...} = {31,61,131,...} with common tail {131,...}. Thus it seems that all terms are leaves of a kind of an inverse prime-tree with branches in A291620 and the root at infinity.

In each equivalence class {s_n(k)} the terms hold: s_n(k+1)-2*s_n(k) >= -1;

(s_n(k+1)+1)/s_n(k) >= 2; lim_{k -> inf} (s_n(k+1)+1)/s_n(k) = 2.

Links

Table of n, a(n) for n=1..57.

Formula

a(n) >= 2*n; a(n) > 10*n - 50; a(n) < 12*n.

a(n) >= e(n) - 1, for n > 1; a(n) < e(n) + n.

Example

For n=1 the Rowland recurrence with e(1)=4 is A084662 with first differences A134734 and records {2,3,5,11,...} gives the least new prime a(1)=2 as the first term of a first equivalence class {2,3,5,11,...} of prime sequences.
For n=2 with e(2)=8 and records {2,7,13,29,59,...} gives the least new prime a(2)=7 as the first term of a second equivalence class {7,13,29,59,...} of prime sequences.
For n=3 with e(3)=16, a(3)=17 the third equivalence class is {17,41,83,167,...}.

Mathematica

For[i = 2; pl = {}; fp = {}, i < 350, i++,
ps = Union@FoldList[Max, 1, Rest@# - Most@#] &@
   FoldList[#1 + GCD[#2, #1] &, 2 i, Range[2, 10^5]];
p = Select[ps, (i <= #) && ! MemberQ[pl, #] &, 1];
If[p != {}, fp = Join[fp, {p}];
  pl = Union[pl,
    Drop[ps, -1 + Position[ps, p[[1]]][[1]][[1]]]]]]; Flatten@fp

Crossrefs

Cf. A291620 (branches), A167168 (equivalence classes), A134734 (first differences of A084662), A134162.

Sequence in context: A317308 A045378 A101568 * A182575 A074884 A227144

Adjacent sequences: A291525 A291526 A291527 * A291529 A291530 A291531

Keyword

nonn

Author

Ralf Steiner, Aug 25 2017

Status

approved