A262324 Conway's triangle of "happy factorizations" (flattened).
0, 0, 1, 1, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 7, 1, 2, 4, 3, 3, 1, 10, 1, 11, 3, 4, 1, 13, 7, 2, 3, 5, 4, 4, 1, 17, 2, 9, 1, 19, 4, 5, 3, 7, 2, 11, 23, 1, 4, 6, 5, 5, 1, 26, 1, 27, 7, 4, 1, 29, 5, 6, 31, 1, 16, 2, 11, 3, 17, 2, 5, 7, 6, 6, 1, 37, 2, 19, 3, 13, 2, 20, 1, 41, 6, 7, 1, 43, 11, 4, 5, 9, 23, 2, 47, 1, 6, 8, 7, 7
Offset: 0
Examples
Triangle begins:
{0,0},
{1,1},
{1,2}, {1,3}, {2,2},
{1,5}, {2,3}, {7,1}, {2,4}, {3,3},
{1,10}, {1,11}, {3,4}, {1,13}, {7,2}, {3,5}, {4,4},
{1,17}, {2,9}, {1,19}, {4,5}, {3,7}, {2,11}, {23,1}, {4,6}, {5,5},
...
The original triangle (adapted and truncated):
...
5^2 ...
4^2 1*26 ...
3^2 1*17 1*27 ...
2^2 1*10 2*9 7*4 ...
1^2 1*5 1*11 1*19 1*29 ...
0^2 1*2 2*3 3*4 4*5 5*6 ...
1^2 1*3 7*1 1*13 3*7 31*1 ...
2^2 2*4 7*2 2*11 16*2 ...
3^2 3*5 23*1 11*3 ...
4^2 4*6 17*2 ...
5^2 5*7 ...
6^2 ...
...
Links
- J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Programs
-
Mathematica
f[0] = {0, 0}; f[32] = {16, 2}(* to speed up *); f[n_] := Do[c = n/b; If[b == c, Return[{b, b}]]; r1 = Reduce[r >= 0 && s >= 0 && c > 1 && b*r^2 + 1 == c*s^2, {r, s}, Integers]; If[r1 =!= False, Return[{b, c}]]; r2 = Reduce[r >= 0 && s >= 0 && r == 2x + 1 && s == 2y + 1 && b*r^2 + 2 == c *s^2, {r, s, x, y}, Integers]; If[r2 =!= False, Return[{b, c}]], {b, Divisors[n]}]; Table[Print["f(", n, ") = ", fn = f[n]]; fn, {n, 0, 49}] // Flatten
Comments