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

1,3

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.

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 *)

Prog

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

Crossrefs

Cf. A007966, A007969, A191855.

Sequence in context: A318762 A262598 A370899 * A129646 A277640 A370906

Adjacent sequences: A191851 A191852 A191853 * A191855 A191856 A191857

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