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A170943 Numbers n with the property that when 1/n is written in base 3 (in either of the two representations, if the representation is ambiguous) the fractional part contains no 1's.

1, 4, 10, 12, 13, 28, 30, 36, 39, 40, 82, 84, 90, 91, 108, 117, 120, 121, 244, 246, 252, 270, 273, 324, 328, 351, 360, 363, 364, 730, 732, 738, 756, 757, 810, 819, 820, 949, 972, 984, 1036, 1053, 1080, 1089, 1092, 1093, 2188, 2190, 2196, 2214, 2268, 2271, 2362, 2430

Offset: 1

J. H. Conway, T. D. Noe and N. J. A. Sloane, Feb 20 2010

That is, neither of the two representations of 1/n in base 3 contain a 1.

This is A121153 without the numbers 3^k, k >= 1. See that entry for further information.

			1/3 in base 3 can be written as either .1 or .0222222... The first version contains a 1, so 3 is not in the sequence.
1/4 in base 3 is .02020202020..., so 4 is in the sequence.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n=1..1144 (terms < 3^21)

Cf. A005823, A005836, A121153, A170951.

A170952 Take the Cantor set sequence A121153 and if the entry m = A121153(n) is in the range 3^k <= m < 3^(k+1), subtract 3^k from it.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 4, 0, 1, 3, 9, 12, 13, 0, 1, 3, 9, 10, 27, 36, 39, 40, 0, 1, 3, 9, 27, 30, 81, 85, 108, 117, 120, 121, 0, 1, 3, 9, 27, 28, 81, 90, 91, 220, 243, 255, 307, 324, 351, 360, 363, 364, 0, 1, 3, 9, 27, 81, 84, 175, 243, 270, 273, 625, 660, 729, 733, 765, 921, 972, 1053

Offset: 1

J. H. Conway, T. D. Noe and N. J. A. Sloane, Feb 22 2010

			If written as a triangle:
0,
0, 1,
0, 1, 3, 4,
0, 1, 3, 9, 12, 13,
0, 1, 3, 9, 10, 27, 36, 39, 40,
0, 1, 3, 9, 27, 30, 81, 85, 108, 117, 120, 121,
0, 1, 3, 9, 27, 28, 81, 90, 91, 220, 243, 255, 307, 324, 351, 360, 363, 364,
0, 1, 3, 9, 27, 81, 84, 175, 243, 270, 273, 625, 660, 729, 733, 765, 921, 972, 1053, 1080, 1089, 1092, 1093,
...

N. J. A. Sloane, Table of n, a(n) for n=1..179

Cf. A121153, A170943, A170944, A170951.

A170944 Complement of A121153.

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 85, 86, 87, 88, 89

Offset: 1

J. H. Conway and N. J. A. Sloane, Feb 20 2010

nonn

If n is a term in this sequence then i/n is not in the Cantor set.

Cf. A121153, A170951.

A170951 Numbers n with the property that some of the fractions i/n (with gcd(i,n)=1, 0 < i/n < 1) are in the Cantor set and some are not.

Original entry on oeis.org

9, 12, 13, 27, 28, 30, 36, 39, 40, 81, 82, 84, 90, 91, 108, 117, 120, 121, 243, 244, 246, 252, 270, 273, 324, 328, 351, 360, 363, 364, 729, 730, 732, 738, 756, 757, 810, 819, 820, 949, 972, 984, 1036, 1053, 1080, 1089, 1092, 1093, 2187

Offset: 1

J. H. Conway and N. J. A. Sloane, Feb 20 2010

Equals A054591 \ {1,3,4,10}.

The natural numbers may be divided into three sets: denominators which force membership in the Cantor set, denominators which deny membership in the Cantor set and denominators which neither force nor deny membership. The first set contains just the numbers 1, 3, 4, 10. The second set is A170944. The third set is the present sequence.

			1/9 is in the Cantor set, but 4/9 is not.

Cf. A005823, A005836, A054591, A121153, A170943, A170944.

A161598 Numbers such that TITO(n) is not equal to n, where TITO(n) = A161594(n).

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 34, 35, 36, 38, 40, 42, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 63, 64, 65, 68, 70, 72, 74, 75, 76, 78, 80, 81, 84, 85, 87, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116

Offset: 1

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

There are no prime numbers in the sequence: A010051(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Complement of A161597.

a161598 n = a161598_list !! (n-1)
a161598_list = filter (\x -> a161594 x /= x) [1..]
-- Reinhard Zumkeller, Oct 14 2011

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Select[Range[200], f[ # ] != # &]

Edited by N. J. A. Sloane, Jun 23 2009

Offset corrected by Reinhard Zumkeller, Oct 14 2011

A161600 Nonprime numbers such that TITO(n) = n, where TITO(n) = A161594(n).

Original entry on oeis.org

1, 4, 6, 8, 9, 22, 26, 33, 39, 44, 46, 55, 62, 66, 69, 77, 82, 86, 88, 93, 99, 121, 143, 169, 187, 202, 206, 226, 242, 252, 253, 262, 286, 299, 303, 309, 339, 341, 343, 363, 393, 404, 422, 446, 451, 466, 473, 482, 484, 505, 525, 583, 606, 616, 622, 626, 633, 662, 669, 671, 682, 686

Offset: 1

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A010051, A161594; subsequence of A161597.

a161600 n = a161600_list !! (n-1)
a161600_list = filter ((== 0) . a010051) a161597_list
-- Reinhard Zumkeller, Oct 14 2011

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k; f[n_] := FromDigits[Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]]; Select[Range[600], f[#] == # && ! PrimeQ[#] &]

is(n)=!isprime(n)&&n==A161594(n) \\ M. F. Hasler, May 11 2015

Edited by N. J. A. Sloane, Jun 23 2009
Offset corrected by Reinhard Zumkeller, Oct 14 2011
Minor edits and more displayed terms from M. F. Hasler, May 11 2015

A161597 Numbers such that TITO(n) = n, where TITO(n) = A161594(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 37, 39, 41, 43, 44, 46, 47, 53, 55, 59, 61, 62, 66, 67, 69, 71, 73, 77, 79, 82, 83, 86, 88, 89, 93, 97, 99, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

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

TITO(p) = p, for any prime p.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Complement of A161598; nonprimes: A161600.

a161597 n = a161597_list !! (n-1)
a161597_list = filter (\x -> a161594 x == x) [1..]
-- Reinhard Zumkeller, Oct 14 2011

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Select[Range[200], f[ # ] == # &]

is(n)={n==A161594(n)} \\ M. F. Hasler, May 11 2015

Edited by N. J. A. Sloane, Jun 23 2009

Offset corrected by Reinhard Zumkeller, Oct 14 2011

A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 13, 41, 51, 61, 17, 81, 19, 2, 12, 22, 23, 42, 52, 26, 72, 82, 29, 3, 31, 23, 33, 241, 53, 63, 37, 281, 39, 4, 41, 24, 43, 44, 54, 46, 47, 84, 94, 5, 312, 421, 53, 45, 55, 65, 372, 481, 59, 6, 61, 62, 36, 46, 551, 66, 67, 482, 69, 7, 71, 27

Offset: 1

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

Might be called TITO(n), turning n inside out then turning outside in.

Here is the operation: take a number n and find its prime factors. Reverse the digits of every prime factor (for example, replace 17 by 71). Multiply the factors respecting multiplicities. For example, if the original number was 17^2*43^3, the new product will be 71^2*34^3. After that, reverse the resulting number.

			a(34) = 241, because 34 = 2*17, f(34) = 2*71 = 142, and reversing gives 241.

M. F. Hasler, Table of n, a(n) for n=1..5000. [From _M. F. Hasler_, Jun 24 2009]
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A161597, A161598, A161600, A071786, A004086, A151764.

a161594 = a004086 . a071786  -- Reinhard Zumkeller, Oct 14 2011

read("transforms") ; A071786 := proc(n) local ifs,a,d ; ifs := ifactors(n)[2] ; a := 1 ; for d in ifs do a := a*digrev(op(1,d))^op(2,d) ; od: a ; end: A161594 := proc(n) digrev(A071786(n)) ; end: seq(A161594(n),n=1..80) ; # R. J. Mathar, Jun 16 2009
# second Maple program:
r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 19 2017

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Table[f[n], {n, 100}]
Table[IntegerReverse[Times@@Flatten[Table[IntegerReverse[#[[1]]],#[[2]]]& /@FactorInteger[n]]],{n,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 21 2016 *)

R=A004086; A161594(n)={n=factor(n);n[,1]=apply(R,n[,1]);R(factorback(n))} \\  M. F. Hasler, Jun 24 2009. Removed code for R here, see A004086 for most recent & efficient version. - M. F. Hasler, May 11 2015

from math import prod
from sympy import factorint
def f(n): return prod(int(str(p)[::-1])**e for p, e in factorint(n).items())
def R(n): return int(str(n)[::-1])
def a(n): return 1 if n == 1 else R(f(n))
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 28 2022

a(p) = p, for prime p.
a(A161598(n)) <> A161598(n); a(A161597(n)) = A161597(n); A010051(a(A161600(n))) = 1.
From M. F. Hasler, Jun 25 2009: (Start)
a( p*10^k ) = p for any prime p.
Proof: if gcd( p, 2*5) = 1, then a( p * 10^k ) = R( R(p) * R(2)^k * R(5)^k ) = R( R(p) * 10^k ) = R(R(p)) = p;
if gcd(p, 2*5) = 2, then p=2 and a( p * 10^k ) = R( R(2)^(k+1) * R(5)^k ) = R( 2 * 10^k ) = 2 = p and mutatis mutandis for gcd(p, 2*5) = 5. (End)

Simpler definition from R. J. Mathar, Jun 16 2009

Edited by N. J. A. Sloane, Jun 23 2009

A161596 Numbers in cycles of RATS sequences.

Original entry on oeis.org

78, 111, 117, 156, 222, 288, 444, 888, 1223, 1677, 3489, 4444, 8888, 11119, 11127, 11667, 11999, 12333, 16777, 23388, 27888, 34589, 44556, 111177, 112333, 228888, 444455, 889999, 1111113, 1177777, 1788899, 2222244, 4446666, 4558889, 11144445, 13444447, 55556688

Offset: 1

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

The set of all numbers in any cycle of RATS sequences, sorted into natural order.
This implies that for any value a(j) in this sequence, A036839(a(j)) is again member of the sequence.
See Branicky link for larger terms. - Michael S. Branicky, Dec 30 2022

			The numbers 111, 222, 444, 888, 1677, 3489, 12333 and 44556 are in the sequence because they are in the cycle shown in A066710. The numbers 117 and 288 are in the cycle demonstrated in A066711.
The numbers 4444, 8888, 16777, 34589, 112333, 444455, ..., 1112278888, 11999, 1119, 1223 are in the cycle started at A161590(4). The numbers 11127 and 23388 are in the cycle started at A161590(7).

Michael S. Branicky, RATS Sequence Cycles.

Cf. A004000, A036839, A066711, A066710, A161590, A161592.

Descriptive comment and examples added by R. J. Mathar, Jul 08 2009

a(20) and beyond from Michael S. Branicky, Dec 30 2022

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

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User: J. H. Conway

A170943 Numbers n with the property that when 1/n is written in base 3 (in either of the two representations, if the representation is ambiguous) the fractional part contains no 1's.

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A170952 Take the Cantor set sequence A121153 and if the entry m = A121153(n) is in the range 3^k <= m < 3^(k+1), subtract 3^k from it.

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A170944 Complement of A121153.

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A170951 Numbers n with the property that some of the fractions i/n (with gcd(i,n)=1, 0 < i/n < 1) are in the Cantor set and some are not.

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A161598 Numbers such that TITO(n) is not equal to n, where TITO(n) = A161594(n).

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A161600 Nonprime numbers such that TITO(n) = n, where TITO(n) = A161594(n).

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A161597 Numbers such that TITO(n) = n, where TITO(n) = A161594(n).

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A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.

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