A191855 Second factor in happy factorization of n-th rectangular number.

2, 5, 3, 10, 4, 13, 2, 17, 9, 5, 7, 11, 26, 4, 29, 6, 3, 2, 37, 19, 13, 41, 7, 4, 9, 2, 50, 13, 53, 27, 5, 8, 19, 58, 4, 61, 2, 65, 33, 17, 3, 14, 9, 73, 74, 4, 11, 3, 82, 28, 85, 43, 89, 10, 4, 31, 2, 5, 97, 2, 101, 51, 21, 106, 4, 109, 11, 37, 16, 113, 57

Offset

1,1

Comments

a(n) > 1;

a(n) = A007967(A007969(n)) = A007969(n) / A191854(n);

(A191854(n), a(n)) is a 1-happy couple;

notation: C 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.

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}]]; A191855 = Reap[Table[Print[n, " ", f[n]];    If[f[n] != {} && f[n] =!= Null, Sow[f[n][[2]]]], {n, 1, 130}]][[2, 1]] (* Jean-François Alcover, Sep 18 2015 *)

Prog

(Haskell)
a191855 = a007967 . a007969  -- Reinhard Zumkeller, Oct 11 2015

Crossrefs

Cf. A007967, A007969, A191854.

Sequence in context: A057337 A163233 A096666 * A064664 A352188 A323637

Adjacent sequences: A191852 A191853 A191854 * A191856 A191857 A191858

Keyword

nonn

Author

Reinhard Zumkeller, Jun 18 2011

Extensions

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

Status

approved