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A191854 First factor in happy factorization of n-th rectangular number.

%I A191854 #20 Feb 28 2020 22:12:07
%S A191854 1,1,2,1,3,1,7,1,2,4,3,2,1,7,1,5,11,17,1,2,3,1,6,11,5,23,1,4,1,2,11,7,
%T A191854 3,1,15,1,31,1,2,4,23,5,8,1,1,19,7,26,1,3,1,2,1,9,23,3,47,19,1,49,1,2,
%U A191854 5,1,27,1,10,3,7,1,2,4,9,2,1,31,1,14,3,1
%N A191854 First factor in happy factorization of n-th rectangular number.
%C A191854 a(n) = A007966(A007969(n)) = A007969(n) / A191855(n);
%C A191854 (a(n), A191855(n)) is a 1-happy couple;
%C A191854 notation: B in the Conway link.
%H A191854 Reinhard Zumkeller, <a href="/A191854/b191854.txt">Table of n, a(n) for n = 1..200</a>
%H A191854 J. H. Conway, <a href="http://www.cs.uwaterloo.ca/journals/JIS/happy.html">On Happy Factorizations</a>, J. Integer Sequences, Vol. 1, 1998, #1.
%t A191854 r[b_, c_] := (red = Reduce[x>0 && y>0 && b*x^2 + 1 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]); f[128] = {}(* to speed up *); f[n_] := f[n] = If[IntegerQ[Sqrt[n]], {}, Do[c = n/b; If[(r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; Return[{b, c, x0, y0}]], {b, Divisors[n] // Most}]]; A191854 = Reap[Table[Print[n, " ", f[n]]; If[f[n] != {} && f[n] =!= Null, Sow[f[n][[1]]]], {n, 1, 130}]][[2, 1]] (* _Jean-François Alcover_, Sep 18 2015 *)
%o A191854 (Haskell)
%o A191854 a191854 = a007966 . a007969  -- _Reinhard Zumkeller_, Oct 11 2015
%Y A191854 Cf. A007966, A007969, A191855.
%K A191854 nonn
%O A191854 1,3
%A A191854 _Reinhard Zumkeller_, Jun 18 2011
%E A191854 Wrong formula removed (thanks to _Wolfdieter Lang_, who pointed this out) by _Reinhard Zumkeller_, Oct 11 2015