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.

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

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 k-ary tree, k > 2, such as a(7) == a(12) == 233.

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).

[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. Müller, Hamiltonian for the zeros of the Riemann zeta function, arXiv:1608.03679 [quant-ph], 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 prime-tree 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 * A356374 A145096 A210498

Adjacent sequences: A291617 A291618 A291619 * A291621 A291622 A291623

Keyword

nonn

Author

Ralf Steiner, Aug 28 2017

Status

approved