cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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

Formula

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

Views

Author

Antonia Münchenbach, Jan 27 2016

Keywords

Comments

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.

Examples

			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).

References

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

Links

Crossrefs

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

Formula

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

Views

Author

Antonia Münchenbach, Jan 28 2016

Keywords

Comments

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.

Examples

			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).

References

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

Crossrefs

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

Formula

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.

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

Original entry on oeis.org

22, 10111, 11112, 100101, 1011122, 2012112, 2021102, 10010122, 10011112, 10101102
Offset: 1

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

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.

Examples

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

References

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

Links

Crossrefs

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

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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

A267496 Autobiographical numbers in base 8: 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, 21322314, 21322315, 21322316, 21322317, 31123314, 31123315, 31123316, 31123317, 31331415, 31331416, 31331417, 31331516, 31331517, 31331617
Offset: 1

Views

Author

Antonia Münchenbach, Jan 16 2016

Keywords

Comments

The T operator numerically summarizes the frequency of digits 0 through 7 in that order when they occur in a number. The numbers and the frequency are written in base 8.
These are all autobiographical numbers in base 8 which lead to a fixed-point or belong to a cycle.
44 numbers are fixed-points. There are 7 cycles with length 2 and three cycles with length 3.

Examples

			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.

References

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

Links

Crossrefs

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

Views

Author

Antonia Münchenbach, Jan 23 2016

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.
Showing 1-9 of 9 results.

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