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.
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
KEYWORD
nonn
AUTHOR
Ralf Steiner, Aug 25 2017
STATUS
approved