A380659 Rectangular array, read by descending antidiagonals: row n shows the numbers whose prime factorization includes n-1 non-Pythagorean primes (including multiplicities).

1, 5, 2, 13, 3, 4, 17, 7, 6, 8, 25, 10, 9, 12, 16, 29, 11, 14, 18, 24, 32, 37, 15, 20, 27, 36, 48, 64, 41, 19, 21, 28, 54, 72, 96, 128, 53, 23, 22, 40, 56, 108, 144, 192, 256, 61, 26, 30, 42, 80, 112, 216, 288, 384, 512, 65, 31, 33, 44, 81, 160, 224, 432, 576, 768, 1024, 73, 34, 38, 60, 84, 162, 320, 448, 864, 1152, 1536, 2048

Offset

1,2

Comments

Every positive integer appears exactly once.

Links

Table of n, a(n) for n=1..78.

Example

Corner:
       1     5    13     17     25     29     37     41     53
       2     3     7     10     11     15     19     23     26
       4     6     9     14     20     21     22     30     33
       8    12    18     27     28     40     42     44     60
      16    24    36     54     56     80     81     84     88
      32    48    72    108    112    160    162    168    176
      64    96   144    216    224    320    324    336    352
     128   192   288    432    448    640    648    672    704
     256   384   576    864    896   1280   1296   1344   1408
     512   768  1152   1728   1792   2560   2592   2688   2816
    1024  1536  2304   3456   3584   5120   5184   5376   5632
    2048  3072  4608   6912   7168  10240  10368  10752  11264
    4096  6144  9216  13824  14336  20480  20736  21504  22528

Mathematica

f[{x_, y_}] := If[Mod[x, 4] == 1, y, -y];
s[n_] := Map[f, FactorInteger[n]];
p[n_] := {Total[Select[s[n], # > 0 &]], -Total[Select[s[n], # < 0 &]]};
t = Table[p[n], {n, 1, 200000}];
u = Map[Last, t];
row[n_] := Flatten[Position[u, n]]
v[n_] := Take[row[n], 12]
Column[Table[v[n], {n, 0, 12}]]
Grid[Table[v[n], {n, 0, 12}]]
w[m_, n_] := row[m][[n]];
Grid[Table[w[m, n], {m, 0, 12}, {n, 1, 12}]]   (* array *)
Table[w[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten  (* sequence *)

Crossrefs

Cf. A000040, A002144, A002145, A083025, A380658.

Sequence in context: A082153 A283944 A294684 * A013946 A261327 A330613

Adjacent sequences: A380656 A380657 A380658 * A380660 A380661 A380662

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 31 2025

Status

approved