A161593
Lengths of new periods in the RATS sequence (0 replacing infinity).
Original entry on oeis.org
0, 8, 2, 18, 2, 2, 2, 14, 2, 3, 2, 2, 2, 6
Offset: 1
a(1)=A114611(0). a(2)=A114611(j=3)=8 with a cycle of length 8 shown in A066710.
A114611(j=6)=8 does not contribute because the cycle is the same as reached from j=3.
a(3)=A114611(9)=2 with a new cycle of length 2 shown in A066711.
A114611(j=12)=8 does not contribute because the cycle is the same as reached from j=3.
A114611(j=15)=8 does not contribute because 15->66->123 is the cycle as reached from j=3.
A114611(j=18)=2 does not contribute because the cycle is the same as reached from j=9.
A114611(j=21)=8 does not contribute because 21->33->66 reaches the same cycle as started from j=3.
a(4)=A114611(j=29)=18.
- Michael S. Branicky, RATS Sequence Cycles.
- Curtis Cooper, RATS.
- Curtis Cooper and Robert E. Kennedy, Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles, Applications of Fibonacci numbers, Vol. 8, Kluwer Acad. Publ., Dordrecht, 1999, pages 83-93.
- Tanya Khovanova, Destinies of Numbers.
A161590
Initial value x of a RATS trajectory x->A036839(x) ending in a cycle unreachable by any smaller initial value.
Original entry on oeis.org
1, 3, 9, 29, 69, 2079, 3999, 6999, 10677, 20169, 10049598, 20008989, 100014888, 100074268
Offset: 1
The RATS (Reverse Add Then Sort) algorithm applied to 69 produces a sequence 69, 156, 78, 156, 78, ...
Its cycle {156, 78} appears not if the algorithm is started with any number in the range 0 to 68, so 69 is added to the sequence.
A161592
Except for the first term the number in the sequence is the smallest number in a new cycle of a RATS sequence with a new destiny. The first term is the best analog of this for the "infinite cycle".
Original entry on oeis.org
12334444, 111, 117, 1223, 78, 111177, 11127, 11144445, 11667, 1111113
Offset: 1
A275218
Numbers in 2-cycles of RATS sequences.
Original entry on oeis.org
78, 117, 156, 288, 11127, 11667, 23388, 27888, 111177, 228888, 111111777, 222888888, 1111122267, 3333337788, 111111117777, 222288888888, 111111111177777, 222228888888888, 111111111111777777, 222222888888888888
Offset: 1
78 is in the sequence because A036839(78) = 156 and A036839(156) = 78.
-
rev:= proc(n) local t,L;
L:= convert(n,base,10);
add(10^j*L[-1-j],j=0..nops(L)-1)
end proc:
sord:= proc(n) local L,t;
L:= sort(convert(n,base,10),`>`);
add(10^j*L[1+j],j=0..nops(L)-1)
end proc:
rats:= proc(n) option remember; sord(n + rev(n)) end proc:
Res:= NULL:
for d from 1 to 15 do
for x1 from 0 to d do
for x2 from 0 to d-x1 do
for x3 from 0 to d-x1-x2 do
for x4 from 0 to d-x1-x2-x3 do
for x5 from 0 to d-x1-x2-x3-x4 do
for x6 from 0 to d-x1-x2-x3-x4-x5 do
for x7 from 0 to d-x1-x2-x3-x4-x5-x6 do
for x8 from 0 to d-x1-x2-x3-x4-x5-x6-x7 do
x9:= d-x1-x2-x3-x4-x5-x6-x7-x8;
L:= [1$x1,2$x2,3$x3,4$x4,5$x5,6$x6,7$x7,8$x8,9$x9];
x:= add(L[-i]*10^(i-1),i=1..d);
if rats(rats(x)) = x then Res:= Res,x fi
od od od od od od od od od:
sort([Res]);
Showing 1-4 of 4 results.
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