OFFSET
2,2
COMMENTS
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)
LINKS
Martin Renner, Table of n, a(n) for n = 2..300
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 8-cycle. Therefore a(10) = 1 + 8 = 9.
MAPLE
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:
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:
seq(NumberOfAttractors(b), b=2..50);
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Martin Renner, Jul 31 2011
STATUS
approved