OFFSET
0,2
COMMENTS
Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.
We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} to achieve codes M -> m for each row n, which is tantamount to ordering according to A059894.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Small plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.
Michael De Vlieger, Enlarged plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.
FORMULA
EXAMPLE
Triangle begins:
1;
2;
6, 3;
30, 10, 15, 5;
210, 42, 70, 14, 105, 21, 35, 7;
...
First terms of this sequence appear bottom to top in the chart below. The
values of n appear in the header, values m = T(n,k) followed
parenthetically by phi(m)/m appear in column n. The x axis plots
according to primepi(gpf(m)), while the y axis plots k according to
phi(m)/m:
0 1 2 3 4
. . . . .
--- 1 ------------------------------------------------
(1/1) . . . .
. . . . .
. . . . .
. . . . 7
. . . 5 (6/7)
. . . (4/5) .
. . . . .
. . . . 35
. . 3 . (24/35)
. . (2/3) . .
. . . . .
. . . . .
. . . . 21
. . . . (4/7)
. . . 15 .
. . . (8/15) .
. 2 . . .
. (1/2) . . .
. . . . .
. . . . 105
. . . . (16/35)
. . . . 14
. . . 10 (3/7)
. . . (2/5) .
. . . . .
. . . . 70
. . 6 . (12/35)
. . (1/3) . .
. . . . 42
. . . 30 (2/7)
. . . (4/15) .
. . . . 210
. . . . (8/35)
...
MATHEMATICA
Prepend[Array[SortBy[#, Last] &@ Map[{#1, #2, EulerPhi[#1]/#1} & @@ {Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits@ #} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 6], {{1, 0, 1}}][[All, All, 1]] // Flatten
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Michael De Vlieger, Apr 13 2019
STATUS
approved