A007968 Type of happy factorization of n.

0, 0, 1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 2, 2

Offset: 0

Author

J. H. Conway

Keywords

• nonn

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.
• Reinhard Zumkeller, Initial Happy Factorization Data for n <= 250

Crossrefs

Cf. A000290, A007969, A007970.

Programs

Haskell

a007968 = (\(hType,,,_,_) -> hType) . h
h 0 = (0, 0, 0, 0, 0)
h x = if a > 0 then (0, a, a, a, a) else h' 1 divs
      where a = a037213 x
            divs = a027750_row x
            h' r []                                = h' (r + 1) divs
            h' r (d:ds)
             | d' > 1 && rest1 == 0 && ss == s ^ 2 = (1, d, d', r, s)
             | rest2 == 0 && odd u && uu == u ^ 2  = (2, d, d', t, u)
             | otherwise                           = h' r ds
             where (ss, rest1) = divMod (d * r ^ 2 + 1) d'
                   (uu, rest2) = divMod (d * t ^ 2 + 2) d'
                   s = a000196 ss; u = a000196 uu; t = 2 * r - 1
                   d' = div x d
hs = map h [0..]
hCouples = map (\(, factor1, factor2, , _) -> (factor1, factor2)) hs
sqrtPair n = genericIndex sqrtPairs (n - 1)
sqrtPairs = map (\(, , _, sqrt1, sqrt2) -> (sqrt1, sqrt2)) hs
-- Reinhard Zumkeller, Oct 11 2015

Formula

a(A000290(n)) = 0; a(A007969(n)) = 1; a(A007970(n)) = 2.