A267496 -id:A267496 - cp's OEIS frontend

A267491 Autobiographical numbers in base 3: numbers which are fixed or belong to a cycle under the operator T.

22, 10111, 11112, 100101, 1011122, 2012112, 2021102, 10010122, 10011112, 10101102

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 2 in that order when they occur in a number. The numbers and the frequency are written in base 3.
These are all autobiographical numbers in base 3 which lead to a fixed-point or belong to a cycle. However, in base 3 there is only one cycle with length 3, all other terms are fixed-points.
10011112, 10101102, 2012112 is a cycle of length 3, which means T(10011112)=10101102, T(10101102)=2012112, T(2012112)=10011112.

			11112 contains four 1's (11 in base 3), and one 2, so T(11112) = 11 1 1 2, and so 11112 is fixed under T.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267493 Autobiographical numbers in base 5: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10313314, 10311233, 21322314, 31123314, 101111314, 101111213, 101111214, 111121314, 1031223314, 10111221314

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 4 in that order when they occur in a number. The numbers and the frequency are written in base 5.

These are all autobiographical numbers in base 5 which lead to a fixed-point or belong to a cycle. However, in base 5 there are no cycles of length greater than 1, that is, all the terms are fixed-points.

			10213223 contains 1 0's, 2 1's, 3 2's and 2 3's.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A047842.

Cf. A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267494 Autobiographical numbers in base 6: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10311233, 10313314, 10313315, 21322314, 21322315, 31123314, 31123315, 31331415, 1031223314, 1031223315, 3122331415, 10111121314, 10111121315, 10111121415, 10111131415, 11112131415, 103142132415, 104122232415, 1011122131415

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 5 in that order when they occur in a number. The numbers and the frequency are written in base 6.
These are all autobiographical numbers in base 6 which lead to a fixed-point or belong to a cycle.
There is one cycle with length 2 (103142132415, 104122232415), all other numbers are fixed-points.

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so
T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267495 Autobiographical numbers in base 7: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10311233, 10313314, 10313315, 10313316, 21322314, 21322315, 21322316, 31123314, 31123315, 31123316, 31331415, 31331416, 31331516, 1031223314, 1031223315, 1031223316, 3122331415, 3122331416, 3122331516, 103142132415, 104122232415, 103142132416, 104122232416, 314213241516, 412223241516, 1011112131415, 1011112131416, 1011112131516, 1011112141516, 1011113141516, 1111213141516, 10414213142516, 10413223241516, 10512223142516, 10512213341516, 101112213141516

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 6 in that order when they occur in a number. The numbers and the frequency are written in base 7.
These are all autobiographical numbers in base 7 which lead to a fixed-point or belong to a cycle.
There are three cycles with length 2 (103142132415 /104122232415, 103142132416/104122232416, 314213241516/412223241516), one cycle with length 3 (10512213341516/10512223142516/10414213142516). 29 numbers are fixed-points.

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so
T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267497 Autobiographical numbers in base 9: numbers which are fixed or belong to a cycle under the operator T (see comments).

Original entry on oeis.org

22, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 21322314, 21322315, 21322316, 21322317, 21322318, 31123314, 31123315, 31123316, 31123317, 31123318

Offset: 1

Antonia Münchenbach, Jan 23 2016

The T operator numerically summarizes the frequency of digits 0 through 8 in that order when if occur in a number. The frequency and the numbers are written in base 9.
These are all autobiographical numbers in base 9 which lead to a fixed-point or belong to a cycle.
68 numbers are fixed-points. There are 15 cycles with length 2 and 6 cycles with length 3.

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Antonia Münchenbach, Table of n, a(n) for n = 1..116
Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.
Antonia Münchenbach, Numbers sorted as fixed-points, cycles with length 2 and 3

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267498 Autobiographical numbers in base 10: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 21322314, 21322315, 21322316, 21322317, 21322318, 21322319, 31123314, 31123315, 31123316, 31123317, 31123318, 31123319, 31331415, 31331416, 31331417, 31331418, 31331419, 31331516, 31331517, 31331518

Offset: 1

Antonia Münchenbach, Jan 25 2016

The T operator numerically summarizes the frequency of digits 0 through 9 in that order when they occur in a number. The numbers and the frequency are written in base 10.
These are all autobiographical numbers in base 10 which lead to a fixed-point or belong to a cycle.
109 numbers are fixed-points. There are 31 cycles with length 2 (62 numbers) and 10 cycles with length 3 (30 numbers).

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Antonia Münchenbach, Table of n, a(n) for n = 1..201
Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.
Antonia Münchenbach, Numbers sorted as fixed-points, cycles with length 2 and 3

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267499 Number of fixed points of autobiographical numbers (A267491 ... A267498) in base n.

Original entry on oeis.org

2, 7, 7, 12, 19, 29, 44, 68, 109, 183

Offset: 2

Antonia Münchenbach, Jan 16 2016

For n>=5, it seems that a(n)=2^(n-4)+1/2*n^2-1/2*n describes the number of fixed points in base n. The formula is correct for 5<=n<=11, but unknown for n>11. We assume it's correct for all n>=5.

			In base two there are only two fixed-points, 111 and 1101001.
In base 3, there are 7 fixed-points: 22, 10111, 11112, 100101, 1011122, 2021102, and 10010122.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016.

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

a(n)=2^(n-4)+1/2*n^2-1/2*n for 5<=n<=11, unknown for n>11.

A267500 Number of fixed points or cycles of autobiographical numbers (A267491 ... A267498) in base n.

Original entry on oeis.org

2, 10, 7, 12, 21, 38, 67, 116, 201, 354

Offset: 2

Antonia Münchenbach, Jan 27 2016

For n>=5, it appears that a(n)=2^(n-3)+2*n^2-17*n+43. This formula is correct for 5<=n<=11, but may not be true for larger n.

			In base two there are only two fixed-points, 111 and 1101001.
In base 3, there are 7 fixed-points: 22, 10111, 11112, 100101, 1011122, 2021102, 10010122 and 1 cycle of length 3 with 2012112, 1010102, 10011112.
In base 10, there are 109 fixed-points, 31 cycles of length 2 (62 numbers) and 10 cycles of length 3 (30 numbers).

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

a(n) = 2^(n-3) + 2*n^2 - 17*n + 43, for 5<=n<=11.

A267502 Number of cycles of length 3 of autobiographical numbers (A267491 ... A267498) in base n.

Original entry on oeis.org

0, 3, 0, 0, 0, 3, 9, 18, 45

Offset: 2

Antonia Münchenbach, Jan 28 2016

a(n) is the number of cycles of length 3 of autobiographical numbers in base n. For n>=5, it seems that a(n)=3/2*n^2-33/2*n+45 describes the number of cycles of length 3 in base n. The formula is correct for 5<=n<=11, but unknown for n>11. We assume it's correct for all n>=5.

			In base two, four, five and six there is no cycle of length 3.
In base three, there is 1 cycle of length 3 with 3 numbers:  10011112, 10101102, 2012112.
In base 10, there are 6 cycles of length 3 (18 numbers).

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267501, A267502.

Conjecture: a(n) = 3/2*n^2 - 33/2*n + 45. The formula is correct for 5<=n<=11, but unknown for n>11. We assume it's correct for all n>=5.

cp's OEIS Frontend

A267491 Autobiographical numbers in base 3: numbers which are fixed or belong to a cycle under the operator T.

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A267493 Autobiographical numbers in base 5: numbers which are fixed or belong to a cycle under the operator T.

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A267494 Autobiographical numbers in base 6: numbers which are fixed or belong to a cycle under the operator T.

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A267495 Autobiographical numbers in base 7: numbers which are fixed or belong to a cycle under the operator T.

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A267497 Autobiographical numbers in base 9: numbers which are fixed or belong to a cycle under the operator T (see comments).

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A267498 Autobiographical numbers in base 10: numbers which are fixed or belong to a cycle under the operator T.

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A267499 Number of fixed points of autobiographical numbers (A267491 ... A267498) in base n.

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A267500 Number of fixed points or cycles of autobiographical numbers (A267491 ... A267498) in base n.

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A267502 Number of cycles of length 3 of autobiographical numbers (A267491 ... A267498) in base n.

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