A288537 -id:A288537 - cp's OEIS frontend

A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2

Offset: 1

Eric W. Weisstein, Dec 16 2005

a(A001651(n)) = 0; a(A114612(n)) = 2; a(A114613(n)) = 3; a(A114614(n)) = 8; a(A114615(n)) = 14; a(A114616(n)) = 18. - Reinhard Zumkeller, Mar 14 2012

Row 10 of A288537. - Andrey Zabolotskiy, Jun 14 2017

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A036839, A161593, A288536, A288537.

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

J. H. Conway and Tanya Khovanova, Jun 14 2009

The values A114611(j) for those starting values j of the RATS mapping x->A036839(x) which end in cycles that cannot be reached starting from any smaller j.
Every integer > 1 appears in this sequence. - Andrey Zabolotskiy, Jun 11 2017
For other terms see Branicky link. - Michael S. Branicky, Dec 30 2022

			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.

Cf. A004000, A036839, A066711, A066710, A288537.

Cf. A161590, A161592, A161596.

Comment and examples added by R. J. Mathar, Jul 07 2009

a(9)-a(14) from Michael S. Branicky, Dec 30 2022

A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 2, 1, 4, 3, 2, 4, 6, 8, 3, 2, 4, 6, 5, 4, 3, 2, 4, 1, 8, 10, 8, 7, 2, 4, 6, 8, 6, 15, 4, 3, 2, 4, 6, 8, 10, 12, 5, 14, 3, 2, 4, 6, 1, 10, 7, 18, 10, 4, 15, 2, 4, 6, 8, 10, 12, 14, 24, 15, 8, 3, 2, 4, 6, 8, 10, 12, 8, 21, 6, 5, 4, 15, 2, 4, 6

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

			17 in base 3 is 122, 122+221=1120->112 which is 14 in decimal, thus RATS(3,17)=14.
The array begins:
1, 3, 3, 3,  3,  3,  7, ...
2, 4, 4, 8,  4,  8,  4, ...
2, 1, 6, 5, 10, 15,  5, ...
2, 4, 6, 8,  6, 12, 18, ...
2, 4, 1, 8, 10,  7, 14, ...

J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839 (row 10), A191780, A288536, A288537.

rats[n_, b_: 10] := FromDigits[Sort[IntegerDigits[n + FromDigits[Reverse[IntegerDigits[n, b]], b], b]], b];
Flatten[Table[rats[n, s + 2 - n], {s, 20}, {n, s}]]

A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

Original entry on oeis.org

1, 3, 2, 2, 8, 4, 3, 2, 0, 28, 90, 8, 72, 3, 4, 2, 64, 0, 18, 4, 18, 20, 396, 8, 160, 120, 18, 6, 28, 4, 5, 2, 210, 384, 240, 0, 648, 1242, 240, 4, 660, 18, 798, 380, 852, 1298, 1771, 8, 0, 160, 16, 372, 520, 1404, 1740, 6, 36, 2072, 1856, 380, 300, 215, 6, 2, 3384, 50, 2310, 3784, 2904

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

Eventual period of 1 under the mapping x->A288535(n,x), or 0 if there is a divergence and thus no eventual period.
Column 1 of A288537.
In Thiel's terms, the zeroes a(10), a(19), and a(37) correspond to quasiperiodic divergent RATS sequences with quasiperiod 2, while a(50)=0 corresponds to a sequence with quasiperiod 3.

			In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so a(3)=3.

S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A114611, A288535, A288537, A237671, A072137.

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A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

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A161593 Lengths of new periods in the RATS sequence (0 replacing infinity).

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A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

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A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

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