A090425 Number of iterations required for happy number A007770(n) to converge to 1.

1, 6, 2, 3, 5, 4, 4, 3, 4, 5, 5, 3, 6, 4, 4, 3, 5, 5, 4, 2, 3, 5, 4, 3, 6, 6, 4, 4, 5, 5, 4, 6, 4, 4, 4, 6, 4, 6, 6, 6, 6, 4, 4, 6, 3, 4, 3, 6, 6, 4, 6, 6, 6, 5, 7, 6, 7, 6, 6, 6, 7, 5, 6, 6, 6, 7, 5, 5, 5, 4, 4, 7, 5, 5, 5, 7, 7, 4, 7, 4, 5, 3, 4, 6, 6, 6, 7, 6, 6, 4, 7, 7, 4, 5, 5, 4, 6, 3, 6, 7, 6, 4

Offset: 1

Author

Eric W. Weisstein, Nov 30 2003

Keywords

• nonn
• base

Comments

The count includes both the start and end.

Examples

			7 is the 2nd happy number and iterated digit squarings and additions give the sequence {7,49,97,130,10,1}, so a(2)=6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• T. Cai and Xia Zhou, On the heights of happy numbers, Rocky Mount. J. Math. 38 (6) (2008) 1921-1926. [From _R. J. Mathar_, Apr 22 2010]
• Eric Weisstein's World of Mathematics, Happy Number.

Crossrefs

Cf. A007770.

Cf. A003132.

Programs

Haskell

a090425 n = snd $ until ((== 1) . fst)
                        (\(u, v) -> (a003132 u, v + 1)) (a007770 n, 1)
-- Reinhard Zumkeller, Aug 07 2012

Mathematica

happy[n_] := If[(list = NestWhileList[Plus @@ (IntegerDigits[#]^2) &, n, UnsameQ, All])[[-1]] == 1, Length[list] - 1, Nothing]; Array[happy, 700] (* Amiram Eldar, Apr 12 2022 *)

Python

from itertools import count, islice
def A090425_gen(): # generator of terms
    for n in count(1):
        c = 1
        while n not in {1,37,58,89,145,42,20,4,16}:
            n = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(n))
            c += 1
        if n == 1:
            yield c
A090425_list = list(islice(A090425_gen(),20)) # Chai Wah Wu, Aug 02 2023