cp's OEIS Frontend

A193583 Number of fixed points under iteration of sum of squares of digits in base b.

%I A193583 #32 Jan 05 2025 19:51:39
%S A193583 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,
%T A193583 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,
%U A193583 3,5,3,7,15,1,7,3,7,3,3,3,7,5,15,1,3,3
%N A193583 Number of fixed points under iteration of sum of squares of digits in base b.
%C A193583 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).
%C A193583 From _Christian N. K. Anderson_, Apr 22 2013: (Start)
%C A193583 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)
%C A193583 From the above chain of logic follows:
%C A193583 - The value of the fixed points can be determined by investigating only 8*A002654(n^2+1) pairs of possibilities.
%C A193583 - a(n) = A000005(n^2+1)-1
%C A193583 - a(n) = A193432(n)-1
%C A193583 a(n)=1 iff n^2+1 is prime, and the value of that single fixed point is 1.
%C A193583 The only odd value of n for which a(n)=9 is n=239.
%C A193583 Several values of a(n) occur very infrequently. For example, a(1068)=13 is the only occurrence of 13 for n < 10000. (End)
%H A193583 Martin Renner and Christian N. K. Anderson, <a href="/A193583/b193583.txt">Table of n, a(n) for n = 2..10000</a> (first 299 values from Martin Renner)
%H A193583 Alan F. Beardon, <a href="http://www.jstor.org/stable/3619884">Sums of Squares of Digits</a>, The Mathematical Gazette,  82(1998), 379-388.
%H A193583 H. G. Grundman, E. A. Teeple, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-5/grundman.pdf">Generalized Happy Numbers</a>, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.
%e A193583 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.
%e A193583 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
%o A193583 (R) library(gmp); y=rep(0,10000)
%o A193583 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
%Y A193583 Equals A193432-1.
%Y A193583 Cf. A007770.
%Y A193583 Cf. A002654, A000005.
%K A193583 nonn,base
%O A193583 2,2
%A A193583 _Martin Renner_, Jul 31 2011