A047841 -id:A047841 - cp's OEIS frontend

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

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.

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

Antonia Münchenbach, Jan 16 2016

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.

			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..67
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.

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.

A108810 Self-describing primes.

Original entry on oeis.org

10153331, 10173133, 10233221, 10311533, 10322321, 12103331, 12163133, 12163331, 12193133, 12311933, 12313319, 15103133, 15233221, 15311633, 15331931, 15333119, 16153133, 16153331, 16173133, 16331531, 16331831, 16333117, 17143331, 17311633, 17331031, 18103133

Offset: 1

G. L. Honaker, Jr., Jul 12 2005

Self-descriptive numbers are read in pairs of digits.

This uses a different method from A047841. Here the digits are described in any order, whereas in A047841 they must be described in increasing order.

			E.g. 10153331 reads "One 0, one 5, three 3's and three 1's", which does indeed describe 10153331.

Computed by Jud McCranie.
Mudge, 'Numbers Count', Personal Computer World, Jun 15 1996

Giovanni Resta, Table of n, a(n) for n = 1..10000
Prime Curios, Self-describing primes

Cf. A047841, A059504, A109775, A109776, A173101.

More terms from Giovanni Resta, Aug 14 2019

A237605 Numbers n such that A047842(n) | n.

Original entry on oeis.org

0, 22, 777, 4444, 303300, 333333, 555555, 588588, 666666, 888688, 2032230, 5055555, 5858558, 6568588, 6868288, 7339393, 8282088, 10213223, 10311233, 10313314, 10313315, 10313316, 10313317, 10313318, 10313319, 20002200, 21322314, 21322315, 21322316, 21322317, 21322318

Offset: 1

Paolo P. Lava, Feb 10 2014

Autobiographical numbers (A047841) are a subset of this sequence.

The first 3 terms which contain more than 9 copies of a digit are 666666666666, 898888888898 and 4444044404444. - Giovanni Resta, Feb 10 2014

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 101 terms from Paolo P. Lava)

Cf. A047841, A047842.

P:=proc(q) local a,b,c,d,f,n,v; print(0); v:=array[0..9];
for n from 1 to q do a:=n; for b from 0 to 9 do v[b]:=0; od;
while a>0 do b:=a mod 10; v[b]:=v[b]+1; a:=trunc(a/10); od; a:=0;
for b from 0 to 9 do if v[b]>0 then c:=10*v[b]+b; f:=0; d:=c;
while d>0 do f:=f+1; d:=trunc(d/10); od; a:=a*10^f+c; fi; od;
if type(n/a,integer) then print(n); fi; od; end: P(10^10);

Select[Range[10^6], Mod[#, FromDigits@ Flatten[IntegerDigits /@ Flatten[ Reverse /@ Tally@ Sort@ IntegerDigits@#]]] == 0 &] (* Giovanni Resta, Feb 10 2014 *)

A047843 Describe n: give frequency of each digit, by increasing size; mention also missing digits between the smallest and largest one.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1011, 21, 1112, 110213, 11020314, 1102030415, 110203040516, 11020304050617, 1102030405060718, 110203040506070819, 100112, 1112, 22, 1213, 120314, 12030415, 1203040516

Offset: 0

N. J. A. Sloane

Other methods to describe or summarize n are: A047842 (as here, but ignoring "missing" digits), A244112 (count digits in order of decreasing size, ignoring missing digits). - M. F. Hasler, Feb 25 2018

			131 contains two 1's, zero 2's and one 3, so a(131) = 210213.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A005151, A047841, A047842.

a[n_] := Module[{T, f}, T = Tally[IntegerDigits[n]]; f[_] = 0; Do[f[t[[1]]] = t[[2]], {t, T}]; Table[{f[k], k}, {k, Min@T[[All, 1]], Max@T[[All, 1]]} ] // Flatten // FromDigits];
a /@ Range[0, 26] (* Jean-François Alcover, Jan 07 2020 *)

A047843(n,S="")={if(n,for(d=vecmin(n=digits(n)),vecmax(n),S=Str(S,#select(t->t==d,n),d));eval(S),10)} \\ M. F. Hasler, Feb 25 2018

More accurate title from M. F. Hasler, Feb 25 2018

cp's OEIS Frontend

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

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A267496 Autobiographical numbers in base 8: 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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A108810 Self-describing primes.

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A237605 Numbers n such that A047842(n) | n.

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