I asked this question on mathstackexchange (https://math.stackexchange.com/questions/3357151/sortingnaturalnumbersbasedonmetric) but I did not get an answer.
I hope that it will go to ask it here:
Let
$$ d ^ J_k (a, b) = 1 frac { sigma_k ( gcd (a, b))} { sigma_k (a) + sigma_k (b) – sigma_k ( gcd (a, b ))} $$ for $ a, b $ two natural numbers $ k $ 0 $.
Then, it is a metric (Jaccard) on natural numbers.
Consider $ d (a, b) = d ^ J (a, b) = sum_ {k = 0} ^ infty frac {1} {2 ^ k} d ^ J_k (a, b) $ and define $ a unlhd b $ Yes Yes $ d (a, 1) the d (b, 1) $
Questions:

Yes $ a unlhd b $ and $ c $ is a natural number, so $ ac unlhd bc $?

Yes $ a unlhd b $ and $ b illhd a $then $ a = b $?

Yes $ a unlhd b $ and $ c unlhd e $then $ ac unlhd be $?

Yes $ gcd (a, c) = gcd (b, c) = $ 1 then $ d (a, b) = d (ca, cb) $. (It's easy to prove, since $ sigma_k $ is multiplicative.)
It seems that the command is lexicographic. I mean, build the vector:
$$ V (a) = (d_0 (a, 1), d_1 (a, 1), d_2 (a, 1) ldots d_k (a, 1) ldots) $$
Then, it seems that $ a unlhd b $ Yes Yes $ V (a) the V (b) $ or $ the $ is the lexicographic order of the given vectors.
Some Sage scripts:
def dd(a,b,k):
return 1sigma(gcd(a,b),k)/(sigma(a,k)+sigma(b,k)sigma(gcd(a,b),k))
def dvec(a,K=10):
return (dd(a,1,k).n() for k in range(K+1))
def d(a,b,K=10):
return sum((1/(2.0**k)*dd(a,b,k) for k in range(K+1)))
sorted(( (d(a,1,10),a) for a in range(1,100)))
((0.000000000000000, 1),
(1.26352353560690, 2),
(1.34364319712592, 3),
(1.40497170796791, 5),
(1.43113204316485, 7),
(1.45520927484247, 11),
(1.46177941791487, 13),
(1.47035738224924, 17),
(1.47331411564775, 19),
(1.47770790490320, 23),
(1.48205464375133, 29),
(1.48313429769773, 31),
(1.48568055948408, 37),
(1.48696820697958, 41),
(1.48752307533606, 43),
(1.48849243175490, 47),
(1.48967436265919, 53),
(1.49061769251944, 59),
(1.49089119792606, 61),
(1.49161439830309, 67),
(1.49202905783214, 71),
(1.49221945289014, 73),
(1.49273313109042, 79),
(1.49303455213995, 83),
(1.49343614594272, 89),
(1.49389469153245, 97),
(1.58038164357392, 4),
(1.62430622092945, 9),
(1.64916894831869, 25),
(1.65681510482431, 49),
(1.70181166355779, 6),
(1.71251996065963, 8),
(1.71920616143290, 10),
(1.72714814044250, 14),
(1.72719187491448, 15),
(1.73288513480220, 21),
(1.73471402834227, 22),
(1.73621232562853, 27),
(1.73681810775224, 26),
(1.73839844837180, 33),
(1.73841153706667, 35),
(1.73958889336088, 34),
(1.73994574914563, 39),
(1.74054992940204, 38),
(1.74198349871905, 46),
(1.74199186845917, 51),
(1.74199942675375, 55),
(1.74270372005527, 57),
(1.74301404059068, 65),
(1.74340792482690, 58),
(1.74376265890517, 62),
(1.74376756472425, 69),
(1.74377384596954, 77),
(1.74436044828411, 85),
(1.74452957378555, 91),
(1.74460070007059, 74),
(1.74482689550047, 87),
(1.74483006361155, 95),
(1.74502525419037, 82),
(1.74509104929573, 93),
(1.74520835591782, 86),
(1.74552845631242, 94),
(1.78213363982812, 16),
(1.79485710740475, 81),
(1.81324507872647, 12),
(1.81896789031508, 18),
(1.81998016814039, 20),
(1.82318502936399, 28),
(1.82423371804470, 32),
(1.82583952513618, 45),
(1.82630550851183, 44),
(1.82690273055893, 50),
(1.82718391487045, 52),
(1.82749364950621, 63),
(1.82828582492105, 75),
(1.82834711474068, 68),
(1.82875220479996, 76),
(1.82912899954912, 99),
(1.82935797699558, 92),
(1.82941227603807, 98),
(1.85218308685392, 64),
(1.86538817693823, 24),
(1.86688267571898, 30),
(1.86835304078615, 40),
(1.86871364331448, 42),
(1.86979510882916, 54),
(1.86979732014398, 56),
(1.87050985618865, 66),
(1.87051243084219, 70),
(1.87101760800471, 78),
(1.87122139031876, 88),
(1.88228470237312, 36),
(1.89491690092618, 48),
(1.89630685447653, 80),
(1.91266763629169, 60),
(1.91309339654707, 72),
(1.91343397284113, 84),
(1.91353207281394, 90),
(1.91368754226931, 96))
# checking numerically if lexicographic sorting:
(x(1) for x in sorted(((dvec(a),a) for a in range(1,100)))) == (x(1) for x in sorted(((d(a,1),a) for a in range(1,100))))
True
Also a related question:
How are the sequences called in a metric space where the distance between two consecutive points converges? (For Cauchy sequences, this distance converges to 0). (We could call them Cauchylike)
I ask this question because it seems that with the metric above, if we have a sequence $ a_1 unlhd a_2 unlhd ldots a_n ldots $ as for each $ n $ there are no natural numbers $ x $ with $ a_n unlhd x unlhd a_ {n + 1} $. We could call such a "complete" sequence. It now seems (numerically) that each "complete" sequence of this type with the given metric resembles that of Cauchy. For example, it seems that prime numbers are complete sequences of Cauchy type.
Of course, if anyone had a proof or glimpse of the above statements, it would be very interesting!