A307544 - cp's OEIS frontend

A307544 Irregular triangle read by rows: T(n,k) = A087207(A307540(n,k)).

0, 1, 3, 2, 7, 5, 6, 4, 15, 11, 13, 9, 14, 10, 12, 8, 31, 23, 27, 19, 29, 21, 25, 30, 17, 22, 26, 18, 28, 20, 24, 16, 63, 47, 55, 59, 39, 43, 51, 61, 35, 45, 53, 57, 37, 62, 41, 49, 46, 54, 33, 58, 38, 42, 50, 60, 34, 44, 52, 56, 36, 40, 48, 32, 127, 95, 111, 119

Offset: 0

Michael De Vlieger, Apr 19 2019

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that 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} (as A059894) to achieve codes M -> m for each row n.
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.
This sequence interprets the code M as a binary value. The sequence is a permutation of the natural numbers since the ratio phi(m)/m is unique for squarefree m.
This sequence and A059894 are identical for 1 <= n <= 23.
Numbers of terms in rows n of this sequence and A059894 (partitioned by powers of 2) that are coincident: 1, 2, 4, 8, 14, 14, 10, 26, 14, 20, 10, 16, 22, 12, 18, 18, 16, 14, 18, 18, 18, 14, 16, ...}.
The graphs of this sequence and A059894 are similar.
The graph of this sequence feature squares of size 2^(j-1) at (x,y) = (h,h) where h = 2^j, integers, that have pi-radian rotational symmetry.

			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. In square brackets, we write the multiplicities of primes in positional order with the smallest prime at right (big-endian). The x axis plots k according to primepi(gpf(m)), while the y axis plots k according to phi(m)/m:
    0       1          2             3             4
    .       .          .             .             .
--- 1 ------------------------------------------------
  (1/1)     .          .             .             .
   [0]      .          .             .             .
    .       .          .             .             .
    .       .          .             .             7
    .       .          .             5           (6/7)
    .       .          .           (4/5)        [1000]
    .       .          .           [100]           .
    .       .          .             .            35
    .       .          3             .          (24/35)
    .       .        (2/3)           .          [1100]
    .       .        [10]            .             .
    .       .          .             .             .
    .       .          .             .            21
    .       .          .             .           (4/7)
    .       .          .            15          [1010]
    .       .          .          (8/15)           .
    .       2          .           [110]           .
    .     (1/2)        .             .             .
    .      [1]         .             .            105
    .       .          .             .          (16/35)
    .       .          .             .          [1110]
    .       .          .             .            14
    .       .          .            10           (3/7)
    .       .          .           (2/5)        [1001]
    .       .          .           [101]           .
    .       .          .             .            70
    .       .          6             .          (12/35)
    .       .        (1/3)           .          [1101]
    .       .        [11]            .            42
    .       .          .            30           (2/7)
    .       .          .          (4/15)        [1011]
    .       .          .           [111]          210
    .       .          .             .           (8/35)
    .       .          .             .          [1111]
...
a(1) = 0 since T(0,1) = 1 = empty product.
a(2) = 1 since T(1,1) = 2 = 2^1 -> binary "1" = decimal 1.
a(3) = 3 since T(2,1) = 6 = 2^1 * 3^1 -> binary "11" = decimal 3.
a(4) = 2 since T(2,2) = 3 = 2^0 * 3^1 -> binary "10" = decimal 2.
a(5) = 7 since T(3,1) = 30 = 2^1 * 3^1 * 5^1 -> binary "111" = decimal 7, etc.
Graph of first 32 terms: (Start)
              x
                       x
                   x
                           x
                 x
                         x
                     x
                x
                             x
                        x
                    x
                            x
                  x
                          x
                      x
                              x
       x
           x
         x
             x
        x
            x
          x
              x
   x
     x
    x
      x
x
  x
x
(End)
From _Antti Karttunen_, Jan 10 2020: (Start)
Arranged as a binary tree:
                                       0
                                       |
                    ...................1...................
                   3                                       2
         7......../ \........5                   6......../ \........4
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    15       11         13       9          14       10         12       8
  31  23   27  19     29  21   25 30      17  22   26  18     28  20   24 16
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Plot comparing A059894 and A307544.
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65535 (including terms 0..16384 previously computed by Michael De Vlieger)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000010, A000040, A000079, A000225, A002110, A005117, A006094, A006530, A007947, A048672, A059894, A067255, A087207, A225679, A225680, A306237, A307540.

Prepend[Array[SortBy[#, Last] &@ Map[{#2, EulerPhi[#1]/#1} & @@ {Times @@  MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits[#, 2]} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 7], {{0, 0, 1}}][[All, All, 1]] // Flatten

up_to = 1023;
rat(n) = { my(m=1, p=2); while(n, if(n%2, m *= (p-1)/p); n >>= 1; p = nextprime(1+p)); (m); };
cmpA307544(a,b) = if(!a,sign(-b),if(!b,sign(a), my(as=logint(a,2), bs=logint(b,2)); if(as!=bs, sign(as-bs), sign(rat(a)-rat(b)))));
A307544list(up_to) = vecsort(vector(1+up_to,n,n-1), cmpA307544);
v307544 = A307544list(up_to);
A307544(n) = v307544[1+n]; \\ Antti Karttunen, Jan 10 2020

For n > 0, row lengths = 2^(n - 1).
T(n,1) = 2^n - 1 = A000225(n).
T(n,2^(n - 1)) = 2^(n - 1).

cp's OEIS Frontend

A307544 Irregular triangle read by rows: T(n,k) = A087207(A307540(n,k)).

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