

A061903


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


7



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



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:


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



KEYWORD

nonn,base,easy


AUTHOR



EXTENSIONS



STATUS

approved



