

A193583


Number of fixed points under iteration of sum of squares of digits in base b.


8



1, 3, 1, 3, 1, 5, 3, 3, 1, 3, 3, 7, 1, 3, 1, 7, 5, 3, 1, 7, 3, 7, 1, 3, 1, 7, 3, 3, 3, 7, 5, 7, 3, 3, 1, 7, 5, 3, 1, 5, 3, 11, 3, 3, 3, 15, 3, 3, 3, 3, 3, 7, 1, 7, 1, 15, 3, 3, 3, 3, 3, 7, 3, 3, 1, 7, 7, 3, 5, 3, 7, 15, 1, 7, 3, 7, 3, 3, 3, 7, 5, 15, 1, 3, 3
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OFFSET

2,2


COMMENTS

If b>=2 and a>=b^2 then S(a,2,b)<a. For each positive integer a, there is a positive integer m such that S^m(a,2,b)<b^2. (Grundman/Teeple, 2001, Lemma 6 and Corollary 7).
From Christian N. K. Anderson, Apr 22 2013: (Start)
It can be shown that no fixed point has more than 2 digits in base b, and that the twodigit number A+Bb must satisfy the condition that (2A1)^2+(2Bb)^2=1+b^2. The number of ways of writing (1+b^2) as the sum of two squares is d(1+b^2)1, where d(n) is the number of divisors of n. (Beardon, 1998, Theorem 3.1)
From the above chain of logic follows:
 The value of the fixed points can be determined by investigating only 8*A002654(n^2+1) pairs of possibilities.
 a(n) = A000005(n^2+1)1
 a(n) = A193432(n)1
a(n)=1 iff n^2+1 is prime, and the value of that single fixed point is 1.
The only odd value of n for which a(n)=9 is n=239.
Several values of a(n) occur very infrequently. For example, a(1068)=13 is the only occurrence of 13 for n < 10000. (End)


LINKS

Martin Renner and Christian N. K. Anderson, Table of n, a(n) for n = 2..10000 (first 299 values from Martin Renner)
Alan F. Beardon, Sums of Squares of Digits, The Mathematical Gazette, 82(1998), 379388.
H. G. Grundman, E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462466.


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.
a(5)=3 because 1 is always a fixed point; also in base 5, decimal 13 > 23 and 2^2+3^2 = 13; decimal 18 > 33 and 3^2+3^2 = 18.  Christian N. K. Anderson, Apr 22 2013


PROG

(R) library(gmp); y=rep(0, 10000)
for(B in 1:10000) y[B]=prod(table(as.numeric(factorize(1+as.bigz(B)^2)))+1)1; y # Christian N. K. Anderson, Apr 22 2013


CROSSREFS

Equals A1934321.
Cf. A007770.
Cf. A002654, A000005.
Sequence in context: A218355 A103790 A249947 * A331731 A309891 A317937
Adjacent sequences: A193580 A193581 A193582 * A193584 A193585 A193586


KEYWORD

nonn,base


AUTHOR

Martin Renner, Jul 31 2011


STATUS

approved



