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

0,4

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

Formula

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

Prog

(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

Crossrefs

Cf. A000290, A007969, A007970.

Sequence in context: A339823 A127506 A353433 * A236532 A077763 A030218

Adjacent sequences: A007965 A007966 A007967 * A007969 A007970 A007971

Keyword

nonn

Author

J. H. Conway

Status

approved