A007967 Second factor in happy factorization of n.

0, 1, 2, 3, 2, 5, 3, 1, 4, 3, 10, 11, 4, 13, 2, 5, 4, 17, 9, 19, 5, 7, 11, 1, 6, 5, 26, 27, 4, 29, 6, 1, 2, 3, 2, 7, 6, 37, 19, 13, 20, 41, 7, 43, 4, 9, 2, 1, 8, 7, 50, 51, 13, 53, 27, 5, 8, 19, 58, 59, 4, 61, 2, 9, 8, 65, 33, 67, 17, 3, 14, 1, 9, 73, 74, 3, 4, 11, 3, 1, 10, 9, 82, 83

Offset

0,3

Comments

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

a(A007969(n)) = A191855(n); a(A007970(n)) = A191857(n). - Reinhard Zumkeller, Jun 18 2011

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..300

J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.

Initial Happy Factorization Data

Mathematica

r[b_, c_,  d_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + d == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, red[[1]], red]); f[n_] := f[n] =  If[IntegerQ[rn = Sqrt[n]], {0, rn, rn, rn, rn},  Catch[Do[b = bc[[1]]; c = bc[[2]]; If[c > 1 && (r0 = r[b, c, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, b, c, x0, y0}]]; If[b > 1 && (r0 = r[c, b, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, c, b, x0, y0}]]; If[(r0 = r[b, c, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, b, c, x0, y0}]]]; If[(r0 = r[c, b, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, c, b, x0, y0}]]]; , {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]}]]]; a[n_] := f[n][[3]]; A007967 = Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 90}] (* Jean-François Alcover, Sep 18 2015 *)

Prog

(Haskell)
import Data.List (genericIndex)
a007967 n = genericIndex a007967_list n
a007967_list = map snd hCouples
-- Pairs hCouples are defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015

Crossrefs

Cf. A191914.

Cf. A007968, A007969, A007970, A191855, A191857.

Sequence in context: A248207 A174621 A369029 * A054494 A306252 A112764

Adjacent sequences: A007964 A007965 A007966 * A007968 A007969 A007970

Keyword

nonn,easy,nice

Author

J. H. Conway

Status

approved