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 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. 1

%I

%S 2,7,17,19,31,43,53,71,67,79,97,103,109,113,127,137,151,163,181,173,

%T 191,197,199,211,229,239,241,251,269,257,271,283,293,317,331,337,349,

%U 367,373,419,409,431,433,439,443,463,491,487,499,523,557,547,577,593,607,599,601

%N 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.

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

%C 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.

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

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

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

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

%e 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.

%e 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.

%e For n=3 with e(3)=16, a(3)=17 the third equivalence class is {17,41,83,167,...}.

%t For[i = 2; pl = {}; fp = {}, i < 350, i++,

%t ps = Union@FoldList[Max, 1, Rest@# - Most@#] &@

%t FoldList[#1 + GCD[#2, #1] &, 2 i, Range[2, 10^5]];

%t p = Select[ps, (i <= #) && ! MemberQ[pl, #] &, 1];

%t If[p != {}, fp = Join[fp, {p}];

%t pl = Union[pl,

%t Drop[ps, -1 + Position[ps, p[]][][]]]]]; Flatten@fp

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

%K nonn

%O 1,1

%A _Ralf Steiner_, Aug 25 2017

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Last modified July 11 15:58 EDT 2020. Contains 335626 sequences. (Running on oeis4.)