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 A262324 Conway's triangle of "happy factorizations" (flattened). 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Conway's triangle is listed by increasing couple products, with duplicate squares removed. LINKS Table of n, a(n) for n=0..99. J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1. EXAMPLE 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 ... ... 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 CROSSREFS Cf. A007966, A007969, A007970, A191854, A191855, A191856. Sequence in context: A336927 A318832 A162348 * A286364 A084216 A347240 Adjacent sequences: A262321 A262322 A262323 * A262325 A262326 A262327 KEYWORD tabf,nonn AUTHOR Jean-François Alcover, Sep 18 2015 STATUS approved

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Last modified March 3 11:19 EST 2024. Contains 370511 sequences. (Running on oeis4.)