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A193594 Number of attractors under iteration of sum of cubes of digits in base b.

%I A193594 #22 Jan 05 2025 19:51:39
%S A193594 1,6,9,6,9,34,11,28,15,46,22,50,49,60,86,86,60,128,22,58,118,93,64,
%T A193594 185,5,109,102,100,122,184,51,94,205,131,173,275,67,216,131,127,34,
%U A193594 360,114,78,215,213,393,479,76,254,634,197,214,496,348,170,437,349,290
%N A193594 Number of attractors under iteration of sum of cubes of digits in base b.
%C A193594 If b>=2 and a >= 2*b^3, then S(a,3,b)<a. For each positive integer a, there is a positive integer m such that S^m(a,3,b)<2*b^3. (Grundman/Teeple, 2001, Lemma 8 and Corollary 9.)
%H A193594 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 A193594 In the decimal system all integers go to (1); (153); (370); (371); (407) or (55, 250,133); (136, 244); (160, 217, 352); (919, 1459) under the iteration of sum of cubes of digits, hence there are five fixed points, two 2-cycles and two 3-cycles. Therefore a(10) = 5 + 2*2 + 2*3 = 15.
%p A193594 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 A193594 NumberOfAttractors:=proc(b) local A,i,Q; A:=[]: for i from 1 to 2*b^3 do Q:=S(i,3,b); A:=[op(A),Q[nops(Q)]]; od: return(nops({op(A)})); end:
%p A193594 seq(NumberOfAttractors(b),b=2..20);
%Y A193594 Cf. A193586.
%K A193594 nonn,base
%O A193594 2,2
%A A193594 _Martin Renner_, Jul 31 2011