A380658 Rectangular array, read by descending antidiagonals: row n shows the numbers whose prime factorization includes n-1 Pythagorean primes (including multiplicities).

2, 3, 1, 4, 5, 25, 6, 10, 50, 125, 7, 13, 65, 250, 625, 8, 15, 75, 325, 1250, 3125, 9, 17, 85, 375, 1625, 6250, 15625

Offset

1,1

Comments

Every positive integer appears exactly once.

Links

Table of n, a(n) for n=1..28.

Example

Corner:
       2      3      4      6      7      8      9     11     12
       1      5     10     13     15     17     20     26     29
      25     50     65     75     85    100    130    145    150
     125    250    325    375    425    500    650    725    750
     625   1250   1625   1875   2125   2500   3250   3625   3750
    3125   6250   8125   9375  10625  12500  16250  18125  18750
   15625  31250  40625  46875  53125  62500  81250  90625  93750

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[First, t];
row[n_] := Flatten[Position[u, n]]
v[n_] := Take[row[n], 12]
Column[Table[v[n], {n, 0, 6}]]
Grid[Table[v[n], {n, 0, 6}]]
w[m_, n_] := row[m][[n]];
Grid[Table[w[m, n], {m, 0, 6}, {n, 1, 12}]]   (* array *)
Table[w[n - k, k], {n, 0, 6}, {k, n, 1, -1}] // Flatten  (* sequence *)

Crossrefs

Cf. A000040, A002144, A002145, A083025, A380659.

Sequence in context: A341130 A330139 A349358 * A046671 A178760 A235715

Adjacent sequences: A380655 A380656 A380657 * A380659 A380660 A380661

Keyword

nonn,tabl,more

Author

Clark Kimberling, Jan 31 2025

Status

approved