

A061903


Number of distinct elements of the iterative cycle: n > sum of digits of n^2.


6



1, 1, 4, 1, 3, 3, 1, 2, 2, 1, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 4, 1, 2, 2, 2, 2, 3, 2, 3, 2, 1, 2, 3, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 1, 3, 3, 3, 3, 2, 2, 3, 3, 2, 1, 4, 1, 2, 2
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OFFSET

0,3


COMMENTS

It seems that any such iterative cycle can contain at most 4 distinct elements.
a(197483417) = 5 is the first counterexample: 136 > 28 > 19 > 10 > 1. In fact this sequence is unbounded, since you can extend any chain leftward with the number k999...999 for suitably chosen k. In particular this gives the (pessimistic) bound that there is some n < 10^21942602 with a(n) = 6.  Charles R Greathouse IV, May 30 2014


LINKS

Robert Israel, Table of n, a(n) for n = 0..10000


EXAMPLE

a(2) = 4 since 2 > 4 > 1+6 = 7 > 4+9 = 13 > 1+6+9 = 16 > 2+5+6 = 13, thus {4,7,13,16} are the distinct elements of the iterative cycle of 2. a(6) = 1 since 6 > 3+6 = 9 > 8+1 = 9 thus 9 is the only element in the iterative cycle of 6.


MAPLE

A:= proc(n) local L, m, x;
L:= {}; x:= n;
do
x:= convert(convert(x^2, base, 10), `+`);
if member(x, L) then return nops(L) fi;
L:= L union {x};
od:
end proc:
seq(A(n), n=0..200); # Robert Israel, May 30 2014


PROG

(PARI) a(n)=my(v=List()); while(1, n=sumdigits(n^2); for(i=1, #v, if(n==v[i], return(#v))); listput(v, n)) \\ Charles R Greathouse IV, May 30 2014


CROSSREFS

Cf. A007953, A004159, A061904A061910.
Sequence in context: A067277 A177951 A222130 * A084118 A046071 A078147
Adjacent sequences: A061900 A061901 A061902 * A061904 A061905 A061906


KEYWORD

nonn,base,easy


AUTHOR

Asher Auel (asher.auel(AT)reed.edu), May 17 2001


EXTENSIONS

Corrected a(0) and example, Robert Israel, May 30 2014


STATUS

approved



