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A262324
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Conway's triangle of "happy factorizations" (flattened).
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0
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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
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OFFSET
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0,6
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COMMENTS
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Conway's triangle is listed by increasing couple products, with duplicate squares removed.
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LINKS
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EXAMPLE
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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 ...
...
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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tabf,nonn
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AUTHOR
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STATUS
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approved
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