A005150 -id:A005150 - cp's OEIS frontend

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, 78, 102, 134, 176, 226, 302, 408, 528, 678, 904, 1182, 1540, 2012, 2606, 3410, 4462, 5808, 7586, 9898, 12884, 16774, 21890, 28528, 37158, 48410, 63138, 82350, 107312, 139984, 182376, 237746, 310036, 403966, 526646, 686646

Offset: 1

Jeffrey Shallit

Row lengths of A034002 and of A220424. - Reinhard Zumkeller, Dec 15 2012

Satisfies a recurrence of order 72. The characteristic polynomial of this recurrence is a degree-72 polynomial that factors as (x-1)*q(x), where q(x) is a degree-71 polynomial. The unique positive real root of q is approximately 1.3036 and is called Conway's constant (A014715), which equals the limiting ratio a(n+1)/a(n). - Nathaniel Johnston, Apr 12 2018 [Corrected by Richard Stanley, Dec 26 2018]

J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Communications, Springer, NY 1987, pp. 173-188.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter J. C. Moses, Table of n, a(n) for n = 1..3000 (first 71 terms from Zak Seidov)
S. R. Finch, Conway's Constant.
S. R. Finch, Conway's Constant. [From the Wayback Machine]
Christoph Koutschan,Regular Languages and their Generating Functions: The Inverse Problem.
Eric Weisstein's World of Mathematics, Look and Say Sequence.

a005341 = length . a034002_row  -- Reinhard Zumkeller, Dec 15 2012

RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ Length[ F[ n ] ], {n, 1, 51} ]
p = {12, -18, 18, -18, 18, -20, -22, 31, 15, -4, -4, -19, 62, -50, -21, -11, 41, 54, -56, -44, 15, -27, -15, 45, -8, 89, -64, -66, -25, 38, 126, -39, -32, -33, -65, 107, 14, 16, -13, -79, 7, 42, 12, 8, -26, -9, 35, -23, -20, -30, 34, 58, -1, -20, -36, -6, 13, 8, 6, 3, -1, -4, -1, -4, -5, -1, 8, 6, 0, -6, -4, 1, 0, 1, 1, 1, 1, -1, -1}; q = {-6, 9, -9, 18, -16, 11, -14, 8, -1, 5, -7, -2, -8, 14, 5, 5, -19, -3, 6, 7, 6, -16, 7, -8, 22, -17, 12, -7, -5, -7, 8, -4, 7, 9, -13, 4, 6, -14, 14, -19, 7, 13, -2, 4, -18, 0, 1, 4, 12, -8, 5, 0, -8, -1, -7, 8, 5, 2, -3, -3, 0, 0, 0, 0, 2, 1, 0, -3, -1, 1, 1, 1, -1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[gf, {x, 0, 99}], x] (* Peter J. C. Moses, Jun 23 2013 *)

print1(a=1);for(i=2,100,print1(",",#Str(a=A005150(2,a))))  \\ M. F. Hasler, Nov 08 2011

a(n) = A055642(A005150(n)) = A055642(A007651(n)). - Reinhard Zumkeller, Dec 15 2012

More terms from Mike Keith

A034002 A005150 expanded into single digits.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 2, 2, 2, 1

Offset: 1

N. J. A. Sloane

			Initial rows                             A005150
  1:  1                                           1
  2:  1,1                                        11
  3:  2,1                                        21
  4:  1,2,1,1                                  1211
  5:  1,1,1,2,2,1                            111221
  6:  3,1,2,2,1,1                            312211
  7:  1,3,1,1,2,2,2,1                      13112221
  8:  1,1,1,3,2,1,3,2,1,1                1113213211
  9:  3,1,1,3,1,2,1,1,1,3,1,2,2,1    31131211131221

Reinhard Zumkeller, Rows n = 1..25 of triangle, flattened
J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188. DOI: 10.1007/978-1-4612-4808-8_53.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 36.
Kevin Watkins, Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem
Kevin Watkins, Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006.
Eric Weisstein's World of Mathematics, Look and Say Sequence
Wikipedia, Look-and-say sequence

See the entry for A005150 for much more about this sequence.
Cf. A088203.
Cf. A005341 (row lengths), A220424 (method B version).

-- see Watkins link, p. 3.
import Data.List (group)
a034002 n k = a034002_tabf !! (n-1) !! (k-1)
a034002_row n = a034002_tabf !! (n-1)
a034002_tabf = iterate
               (concat . map (\xs -> [length xs, head xs]) . group) [1]
-- Reinhard Zumkeller, Aug 09 2012

from sympy import flatten
l=[1]
L=[1]
n=s=1
y=''
while n<21:
    x=str(l[n - 1]) + ' '
    for i in range(len(x) - 1):
        if x[i]==x[i + 1]: s+=1
        else:
            y+=str(s)+str(x[i])
            s=1
    x=''
    n+=1
    l.append(int(y))
    L.append([int(a) for a in list(y)])
    y=''
    s=1
print(l) # A005150
print(flatten(L)) # Indranil Ghosh, Jul 05 2017

A005150(n) = Sum_{k=1..A005341(n)} T(n,k)*10^(A005341(n) - k). - Reinhard Zumkeller, Dec 15 2012

Offset changed and keyword tabf added by Reinhard Zumkeller, Aug 09 2012

A079562 Number of distinct prime factors of n-th term of Look and Say Sequence A005150.

Original entry on oeis.org

0, 1, 2, 2, 2, 1, 1, 4, 2, 2, 4, 3, 3, 5, 7, 7, 7, 5

Offset: 1

Joseph L. Pe, Jan 25 2003

a(16) >= 5. - Nathaniel Johnston, Nov 02 2010

a(18) >= 5. - Giovanni Resta, May 20 2020

Andy Huchala, Factors of A005150(n)

Cf. A001221, A005150, A334132.

s[1]=1; s[n_] := s[n] = FromDigits[ Flatten[{ IntegerDigits@ Length@ #, First@ #} & /@ Split[ IntegerDigits@ s[n-1] ]]]; PrimeNu /@ s /@ Range[15] (* Giovanni Resta, May 20 2020 *)

a(n) = A001221(A005150(n)). - Michel Marcus, Apr 09 2022

a(11)-a(15) from Nathaniel Johnston, Nov 02 2010
a(16)-a(17) from Giovanni Resta, May 20 2020
a(18) from Andy Huchala, Apr 08 2022

A004977 Sum of digits of n-th term in Look and Say sequence A005150.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 13, 16, 23, 32, 44, 56, 76, 102, 132, 174, 227, 296, 383, 505, 679, 892, 1151, 1516, 1988, 2602, 3400, 4410, 5759, 7519, 9809, 12810, 16710, 21758, 28356, 36955, 48189, 62805, 81803, 106647, 139088, 181301, 236453, 308150, 401689

Offset: 1

Clark Kimberling

It appears that the ratio of consecutive terms approaches Conway's constant 1.303.. (A014715). The terms divided by the numbers of added digits also would tend to a constant, i.e. A004977(n)/A005341(n)->const. If the digits in A005150 occur with constant probabilities c1, c2, c3 then A004977(n)=A005341(n)*(c1+2*c2+3*c3) and this conjecture entails the convergences noted here. - Alexandre Losev, Aug 31 2005

Peter J. C. Moses, Table of n, a(n) for n = 1..1000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 9
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

Cf. A005150.

Cf. A005150, A005341, A014715.

RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ Apply[ Plus, F[ n ] ], {n, 1, 51} ]
p={-4,8,-7,-10,15,18,11,-65,-4,27,7,9,-62,47,56,-32,-46,-8,67,44,-16,24,2,-59,-20,-65,84,122,-51,-38,-131,10,91,24,39,-89,-42,39,12,45,-40,-63,39,40,10,-19,-58,47,51,-7,-43,-67,32,41,20,-13,-24,-3,8,0,0,0,0,10,5,-3,-11,-6,5,7,3,-2,-1,-1,-1,-1,0,1,1}; q={6,-9,9,-18,16,-11,14,-8,1,-5,7,2,8,-14,-5,-5,19,3,-6,-7,-6,16,-7,8,-22,17,-12,7,5,7,-8,4,-7,-9,13,-4,-6,14,-14,19,-7,-13,2,-4,18,0,-1,-4,-12,8,-5,0,8,1,7,-8,-5,-2,3,3,0,0,0,0,-2,-1,0,3,1,-1,-1,-1,1}; gf=Fold[x #1+#2&,0,p]/Fold[x #1+#2&,0,q]; CoefficientList[Series[gf,{x,0,99}],x] (* Peter J. C. Moses, Jun 24 2013 *)

A087282 An infinite audioactive word, one of three in the cycle that results from the limit of the 'Look and Say' sequence using method A with an initial term of 1 (A005150).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2

Offset: 1

Paul D. Hanna, Aug 28 2003

Performing 'Look and Say' once generates A087283, twice results in A087284, while three operations yield the original infinite word.

Cf. A005150, A087283, A087284.

A119566 Periodic table of elements associated with the Look-and-Say sequence A005150.

Original entry on oeis.org

22, 13112221133211322112211213322112, 312211322212221121123222112, 111312211312113221133211322112211213322112, 1321132122211322212221121123222112, 3113112211322112211213322112

Offset: 1

N. J. A. Sloane, May 31 2006

See A005150 for further comments, links and references.

J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.
J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.

Henry Bottomley, Table of n, a(n) for n = 1..92 (with a(41) corrected by Shashank Bhatt).
Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements

Cf. A005150, A213979 (in lexicographic order), A215403 (transuranic isotopes).

A334132 Smallest prime factor of n-th term in Look and Say sequence A005150, with a(1)=1.

Original entry on oeis.org

1, 11, 3, 7, 11, 312211, 13112221, 11, 5581, 26966089, 7, 20328937, 29, 3, 3, 3, 1637, 103, 50593, 43, 13, 19, 17, 103, 31, 19, 7, 3, 19, 1208033, 23, 3, 3, 83, 3, 233, 3, 3

Offset: 1

Bernard Schott, Apr 15 2020

The terms in A005150 that are known to be primes are a(2) = 11, a(6) = 312211 and a(7) = 13112221 (A100108).
a(n) = 3 iff A004977(n) is positive and divisible by 3.
a(39) > 2*10^9. a(62) > 10^7. - Tyler Busby, Jan 25 2023

			A005150(7) = 13112221 is prime and a(7) = 13112221.
A005150(9) = 31131211131221 = 5581 * 5578070441, hence a(9) = 5581.

Tyler Busby, Table of n, a(n) for n = 1..75 with -1 for those entries where a(n) has not yet been found, Jan 25 2023.
Wikipedia, Look-and-say sequence

Cf. A004977 (sum of digits of terms of A005150), A005150 (Look and Say sequence), A020639, A079562, A100108 (primes in A005150).

a(n) = A020639(A005150(n)).

a(17)-a(38) from Jinyuan Wang, Apr 15 2020

A037033 Earliest sequence of 6 primes according to the rules stipulated in A005150.

Original entry on oeis.org

233, 1223, 112213, 21221113, 1211223113, 11122122132113

Offset: 0

Carlos Rivera

Carlos Rivera, Puzzle 36. Sequences of "descriptive primes", The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 999. In Memoriam to John Horton Conway, The Prime Puzzles and Problems Connection.

Cf. A005150, A038131, A038132.

Name edited by Michel Marcus, May 09 2020

A038131 Second earliest sequence of 6 primes according to the rules stipulated in A005150.

Original entry on oeis.org

120777781, 111210471811, 311211101417111821, 13211231101114111731181211, 111312211213211031143117132118111221, 31131122211211131221101321141321171113122118312211

Offset: 1

Carlos Rivera

Carlos Rivera, Puzzle 36. Sequences of "descriptive primes", The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 999. In Memoriam to John Horton Conway, The Prime Puzzles and Problems Connection.

Cf. A005150, A037033, A038132.

Name edited by Michel Marcus, May 09 2020

A038132 Third earliest sequence of 6 primes according to the rules stipulated in A005150.

Original entry on oeis.org

402266411, 141022261421, 11141110321611141211, 31143110131211163114111221, 132114132110111311123116132114312211, 1113122114111312211031133112132116111312211413112221

Offset: 1

Carlos Rivera

Carlos Rivera, Puzzle 36. Sequences of "descriptive primes", The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 999. In Memoriam to John Horton Conway, The Prime Puzzles and Problems Connection.

Cf. A005150, A037033, A038131.

Name edited by Michel Marcus, May 09 2020

cp's OEIS Frontend

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

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A034002 A005150 expanded into single digits.

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A079562 Number of distinct prime factors of n-th term of Look and Say Sequence A005150.

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Mathematica

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A004977 Sum of digits of n-th term in Look and Say sequence A005150.

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Mathematica

A087282 An infinite audioactive word, one of three in the cycle that results from the limit of the 'Look and Say' sequence using method A with an initial term of 1 (A005150).

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Comments

Crossrefs

A119566 Periodic table of elements associated with the Look-and-Say sequence A005150.

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Author

Keywords

Comments

References

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Crossrefs

A334132 Smallest prime factor of n-th term in Look and Say sequence A005150, with a(1)=1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A037033 Earliest sequence of 6 primes according to the rules stipulated in A005150.

Views

Author

Keywords

Links

Crossrefs

Extensions

A038131 Second earliest sequence of 6 primes according to the rules stipulated in A005150.

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Author