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
Keywords
Examples
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.
Links
- Carl M. Bender, Dorje C. Brody, Markus P. Müller, Hamiltonian for the zeros of the Riemann zeta function, arXiv:1608.03679 [quant-ph], 2016.
Programs
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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
Formula
a(n) > A291528(n) || a(n) == 0.
Comments