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

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, 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, 5, 1, 27, 1, 10, 3, 7, 1, 2, 4, 9, 2, 1, 31, 1, 14, 3, 1

Offset: 1

Author

Reinhard Zumkeller, Jun 18 2011

Keywords

• nonn

Comments

a(n) = A007966(A007969(n)) = A007969(n) / A191855(n);

(a(n), A191855(n)) is a 1-happy couple;

notation: B in the Conway link.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..200
• J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.

Crossrefs

Cf. A007966, A007969, A191855.

Programs

Haskell

a191854 = a007966 . a007969  -- Reinhard Zumkeller, Oct 11 2015

Mathematica

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 *)

Extensions

Wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015