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A291620 Branch term s_n(b), b > 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 start values >= 4.

%I A291620 #29 Oct 30 2017 12:54:24
%S A291620 0,0,0,0,131,0,233,167,2381,647,0,233,0,941,263,0,0,353,0,0,797,0,0,0,
%T A291620 941,0,0,8273,569,0,0,569,1181,0,0,22133,761,0,761,1721,839,1811,881,
%U A291620 0,1811,929,1973,0,0,1049,1181,9323,2309,1187,0,2441,2441
%N A291620 Branch term s_n(b), b > 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 start values >= 4.
%C A291620 See A291528 (leaves) for equivalence classes.
%C A291620 If the conjecture of an inverse tree of primes with the leaves in A291528 using the same index n holds, except a(1)=0 all terms a(n) == 0 indicates that the branch point is not yet found.
%C A291620 This is a k-ary tree, k > 2, such as a(7) == a(12) == 233.
%C A291620 Maybe these simple Rowland-like recurrences giving all primes are related to a simple bounded physical quantum system with a "Hamiltonian for the zeros of the Riemann zeta function" (cf. Bender et al.) having degenerated energy eigenvalues a(n).
%C A291620 [Note: the editors feel that any such connection is extremely unlikely. - _N. J. A. Sloane_, Oct 30 2017]
%H A291620 Carl M. Bender, Dorje C. Brody, Markus P. Müller, <a href="https://arxiv.org/abs/1608.03679">Hamiltonian for the zeros of the Riemann zeta function</a>, arXiv:1608.03679 [quant-ph], 2016.
%F A291620 a(n) > A291528(n) || a(n) == 0.
%e A291620 n=5: Some equivalence classes of prime sequences {s_n(k)} have the same tail for a constant C_n < k, such as {s_2(k)} = {7,13,29,59,131,...} and {s_5(k)} = {31,61,131,...} with common tail {a(5),...} = {131,...} and the branch 131 = a(5). Thus it seems that all terms != 0 are branches of a kind of an inverse prime-tree with the root at infinity.
%t A291620 For[i = 2; pl = {}; fp = {}; bp = {}, i < 350, i++,
%t A291620 ps = Union@FoldList[Max, 1, Rest@# - Most@#] &@
%t A291620    FoldList[#1 + GCD[#2, #1] &, 2 i, Range[2, 10^5]];
%t A291620 p = Select[ps, (i <= #) && ! MemberQ[pl, #] &, 1];
%t A291620 If[p != {},
%t A291620   fp = Join[fp, {p}];
%t A291620   b = Select[Drop[ps, po = Position[ps, p[[1]]][[1]][[1]]],
%t A291620     MemberQ[pl, #] &, 1];
%t A291620   If[b != {}, bp = Join[bp, {b}], bp = Join[bp, {{0}}]];
%t A291620   pl = Union[pl, Drop[ps, po - 1]]]]; Flatten@bp
%Y A291620 Cf. A134162, A134734, A167168 (equivalence classes), A291528 (leaves)
%K A291620 nonn
%O A291620 1,5
%A A291620 _Ralf Steiner_, Aug 28 2017