login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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 two-digit number A+Bb must satisfy the condition that (2A-1)^2+(2B-b)^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), 379-388.

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.

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 A193432-1.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 28 10:29 EDT 2021. Contains 346326 sequences. (Running on oeis4.)