A193585 Number of cycles under iteration of sum of squares of digits in base b.

0, 1, 0, 1, 1, 2, 3, 2, 1, 2, 4, 3, 2, 7, 1, 2, 1, 3, 1, 6, 2, 8, 4, 6, 1, 5, 4, 6, 2, 8, 6, 5, 3, 5, 4, 5, 3, 6, 1, 7, 6, 6, 2, 5, 4, 11, 4, 4, 4, 6, 3, 11, 4, 9, 4, 8, 4, 6, 6, 5, 4, 9, 6, 5, 2, 6, 3, 7, 7, 8, 5, 14, 5, 8, 3, 6, 3, 4, 5, 10, 5, 10, 6, 8, 5

Offset

2,6

Comments

If b>=2 and a>=b^2 then S(a,2,b)<a. For each positive integer a, there is an positive integer m such that S^m(a,2,b)<b^2. (Grundman/Teeple, 2001, Lemma 6 and Corollary 7).

Links

Martin Renner, Table of n, a(n) for n = 2..300

H. G. Grundman, E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.

Example

In the decimal system all integers go to (1) or (4, 16, 37, 58, 89, 145, 42, 20) under the iteration of sum of squares of digits, hence there is one fixed point and one cycle. Therefore a(10) = 1.

Crossrefs

Cf. A193583, A193586.

Sequence in context: A114409 A340680 A343897 * A245436 A285581 A222173

Adjacent sequences: A193582 A193583 A193584 * A193586 A193587 A193588

Keyword

nonn,base

Author

Martin Renner, Jul 31 2011

Status

approved