OFFSET
1,1
COMMENTS
The fixed points are (0,0) and (16,16) (i.e., if x(0)=16 and x(1)=16 then all x(n)=16). Cycles include (23, 32, 33, 24), (19, 28, 30, 21, 20), and (23, 34, 42, 31, 43, 74, 82, 82, 86, 88, 62, 50, 45).
Are there other cycles? Is T(i,j) ever -1? For 1 <= i <= 3000 and 1 <= j <= 3000, T(i,j) is never -1 and no other cycles are encountered.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)
EXAMPLE
Table begins
2, 20, 19, 15, 10, 10, 8, 14, 14, 8, 13, 8, ...
21, 19, 18, 18, 9, 9, 15, 14, 14, 15, 12, 15, ...
20, 10, 18, 14, 17, 17, 8, 14, 14, 8, 8, 8, ...
16, 10, 9, 14, 17, 17, 13, 13, 13, 13, 12, 13, ...
11, 16, 15, 14, 9, 9, 8, 16, 16, 8, 12, 8, ...
11, 16, 15, 14, 9, 9, 8, 16, 16, 8, 12, 8, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
11, 15, 15, 14, 13, 13, 12, 13, 13, 12, 15, 12, ...
11, 15, 15, 14, 13, 13, 12, 13, 13, 12, 15, 12, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
14, 7, 6, 6, 23, 23, 4, 16, 16, 4, 12, 4, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
T(1,7) = 8 because starting at x(0)=1, x(1)=7 we have x(2)=7, x(3)=14, x(4)=16, x(5)=17, x(6)=25, x(7)=27, x(8)=19, x(9)=28, and (19,28) is in the cycle (19, 28, 30, 21, 20).
MAPLE
spf:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
Cyc:= {[0, 0], [16, 16], [32, 33], [33, 24], [24, 23], [23, 32], [28, 30], [30, 21], [21, 20], [20, 19], [19, 28], [34, 42], [42, 31], [31, 43], [43, 74], [74, 82], [82, 82], [82, 86], [86, 88], [88, 62], [62, 50], [50, 45], [45, 23], [23, 34]}:
f:= proc(t) local count, x;
count:= 0;
x:= t;
while count < 1000 do
if member(x, Cyc) then return count fi;
x:= [x[2], spf(x[1])+spf(x[2])];
count:= count+1;
od;
FAIL
end proc:
seq(seq(f([i, k-i]), i=1..k-1), k=2..14);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
J. M. Bergot and Robert Israel, Nov 17 2020
STATUS
approved