

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 Rowlandlike recurrences with increasing even start values >= 4.


1



0, 0, 0, 0, 131, 0, 233, 167, 2381, 647, 0, 233, 0, 941, 263, 0, 0, 353, 0, 0, 797, 0, 0, 0, 941, 0, 0, 8273, 569, 0, 0, 569, 1181, 0, 0, 22133, 761, 0, 761, 1721, 839, 1811, 881, 0, 1811, 929, 1973, 0, 0, 1049, 1181, 9323, 2309, 1187, 0, 2441, 2441
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OFFSET

1,5


COMMENTS

See A291528 (leaves) for equivalence classes.
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.
This is a kary tree, k > 2, such as a(7) == a(12) == 233.
Maybe these simple Rowlandlike 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).
[Note: the editors feel that any such connection is extremely unlikely.  N. J. A. Sloane, Oct 30 2017]


LINKS

Table of n, a(n) for n=1..57.
Carl M. Bender, Dorje C. Brody, Markus P. MÃ¼ller, Hamiltonian for the zeros of the Riemann zeta function, arXiv:1608.03679 [quantph], 2016.


FORMULA

a(n) > A291528(n)  a(n) == 0.


EXAMPLE

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 primetree with the root at infinity.


MATHEMATICA

For[i = 2; pl = {}; fp = {}; bp = {}, 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}];
b = Select[Drop[ps, po = Position[ps, p[[1]]][[1]][[1]]],
MemberQ[pl, #] &, 1];
If[b != {}, bp = Join[bp, {b}], bp = Join[bp, {{0}}]];
pl = Union[pl, Drop[ps, po  1]]]]; Flatten@bp


CROSSREFS

Cf. A134162, A134734, A167168 (equivalence classes), A291528 (leaves)
Sequence in context: A184283 A246242 A001331 * A145096 A210498 A050671
Adjacent sequences: A291617 A291618 A291619 * A291621 A291622 A291623


KEYWORD

nonn


AUTHOR

Ralf Steiner, Aug 28 2017


STATUS

approved



