cp's OEIS Frontend

A193586 Number of attractors under iteration of sum of squares of digits in base n.

%I A193586 #22 Jan 05 2025 19:51:39
%S A193586 1,5,1,6,9,13,10,8,9,9,20,13,12,35,7,15,7,21,27,37,24,36,32,26,10,36,
%T A193586 27,28,10,56,22,26,23,63,39,27,19,67,9,36,40,54,54,48,18,73,52,75,18,
%U A193586 117,52,74,22,65,48,53,45,44,43,18,30,67,39,49,87,111,15
%N A193586 Number of attractors under iteration of sum of squares of digits in base n.
%C A193586 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)
%H A193586 Martin Renner, <a href="/A193586/b193586.txt">Table of n, a(n) for n = 2..300</a>
%H A193586 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 A193586 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 8-cycle. Therefore a(10) = 1 + 8 = 9.
%p A193586 S:=proc(n,p,b) local Q,k,N,z; Q:=[convert(n,base,b)]; for k from 1 do N:=Q[k]; z:=convert(sum(N['i']^p,'i'=1..nops(N)),base,b); if not member(z,Q) then Q:=[op(Q),z]; else Q:=[op(Q),z]; break; fi; od; return Q; end:
%p A193586 NumberOfAttractors:=proc(b) local A,i,Q; A:=[]: for i from 1 to b^2 do Q:=S(i,2,b); A:=[op(A),Q[nops(Q)]]; od: return(nops({op(A)})); end:
%p A193586 seq(NumberOfAttractors(b),b=2..50);
%Y A193586 Cf. A193583, A193585.
%K A193586 nonn,base
%O A193586 2,2
%A A193586 _Martin Renner_, Jul 31 2011