

A191854


First factor in happy factorization of nth rectangular number.


8



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
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OFFSET

1,3


COMMENTS

a(n) = A007966(A007969(n)) = A007969(n) / A191855(n);
(a(n), A191855(n)) is a 1happy couple;
notation: B in the Conway link.


LINKS

Table of n, a(n) for n=1..80.
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.


FORMULA

a(n) * A191860(n)^2 + 1 = A191855(n) * A191861(n)^2.


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]] (* JeanFrançois Alcover, Sep 18 2015 *)


CROSSREFS

Sequence in context: A205858 A053222 A262598 * A129646 A165401 A213074
Adjacent sequences: A191851 A191852 A191853 * A191855 A191856 A191857


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Jun 18 2011


STATUS

approved



