A045918 -id:A045918 - cp's OEIS frontend

A049190 Start with 1. Convert to base 2, describe it in base 2, convert to base 10. Repeat with the new result.

1, 3, 5, 59, 245, 2491, 235253, 127756731, 330567489269, 258479716298484155, 36823182192123209878050549, 25576412117054296344209353299113896379, 10994511204169842163496446583221775727830456269734123253

Offset: 1

Olivier Gérard

a(n) is A001387(n) converted to base 10. - Nathan Fox, Mar 07 2018

"Describe" means to apply the "look-and-say" function (cf. A045918), but the "count" is again expressed in binary (and concatenated with the digit), cf. examples. - M. F. Hasler, Jul 12 2025

			1 -> one 1 -> 11 -> 3;
3 -> 11 -> two 1s -> 101 -> 5;
5 -> 101 -> one 1, one 0, one 1 -> 111011 -> 59;
etc.

Cf. A005150, A001387.

A049190_first(N=13)=vector(N, i, N=if(i>1, my(d=binary(N), j=0); d=concat(d[^1]-d[^-1],-1); fromdigits(concat([concat(binary(-j+j=n), d[n]<0) | n<-[1..#d], d[n]]), 2), 1)) \\ M. F. Hasler, Jul 12 2025

Definition and Example corrected by Nathan Fox, Mar 07 2018

A045981 Describe all the previous terms!.

Original entry on oeis.org

1, 11, 1121, 1121211211, 112121121121121112211221, 112121121121121112211221211211122112211221123122212211

Offset: 1

Anthony Joseph Splendora

Previous terms are described one by one in succession. Next terms are 118, 256, 564 and 1262 digits long.

Reinhard Zumkeller, Table of n, a(n) for n = 1..9
Eric Weisstein's World of Mathematics, Look and Say Sequence
Wikipedia, Look-and-say sequence

Cf. A005150, A045982.

Cf. A045918.

a045981 n = a045981_list !! (n-1)
a045981_list = 1 : f 1 [] where
   f x zs = y : f y zs' where
     y = read (concatMap show zs')
     zs' = zs ++ [a045918 x]
-- Reinhard Zumkeller, Feb 28 2014

More terms from Patrick De Geest, Jun 15 1999

A260521 Concatenate the positions of digits 9, 8,..., 0 in the decimal representation of n, using 1 for the rightmost digit etc., and 0 when the digit does not occur.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 21, 120, 120, 1020, 10020, 100020, 1000020, 10000020, 100000020, 1000000020, 201, 210, 1200, 1200, 10200, 100200, 1000200, 10000200, 100000200, 1000000200

Offset: 0

M. F. Hasler, Jul 28 2015

			Consider n=1103, digits '9' through '4' do not occur, digit '3' at 1st position (looking from the right), digit '2' does not occur, digit '1' in 3rd and 4th place, and digit '0' is at 2nd position. Thus a(1103) = 00000010342 = 10342.

Cf. A045918 (the classical "look and say" sequence).

Cf. A260519 - A260522 for variants (counting from the right or starting with digit 0).

A260521=a(n)={my(p=vector(10));for(i=1,#n=if(n,Vecrev(digits(n)),[0]),p[n[i]+1]=if(p[n[i]+1],concat(p[n[i]+1],i),i));p=concat(Vecrev(p));p[1]=Str(p[1]);eval(concat(p))}

A260529 Concatenate the positions of digits 9, 8,..., 0 in the decimal representation of n, using 1 for the rightmost digit etc., skip it if a digit does not occur.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 12, 12, 12, 12, 12, 12, 12, 12, 12, 21, 21, 12, 12, 12, 12, 12, 12, 12, 12, 21, 21, 21, 12, 12, 12, 12, 12, 12, 12, 21, 21, 21, 21, 12, 12, 12, 12, 12, 12, 21, 21, 21, 21, 21, 12, 12, 12, 12, 12, 21, 21, 21, 21, 21, 21, 12

Offset: 0

M. F. Hasler, Jul 28 2015

Sequence A260275 lists fixed points of this function.

			Consider n=1103, digits 4-9 do not occur, the digit '3' is at the 1st position (looking from the right), digit '2' does not occur, digit '1' is in 3rd and 4th place, digit '0' at 2nd position. Thus a(1103) = 1342.

Cf. A260526 - A260528 and A260519 - A260522 for variants; A045918 for the classical "look and say" sequence.

a(n)={my(p=vector(10));for(i=1,#n=if(n,Vecrev(digits(n)),[0]),p[n[i]+1]=if(p[n[i]+1],concat(p[n[i]+1],i),i));if(1<#p=concat(Vecrev(select(t->#t,p))),p[1]=Str(p[1]);eval(concat(p)),p)}

A329818 Look left and tell the least frequent digit string; a(0) = 0, a(n) = concat(c,s), where c > 0 is the number of times the string s, composed of one or more digits, has appeared in a(0) to a(n-1) such that a(n) is minimal. Digit strings with leading zeros are ignored.

Original entry on oeis.org

0, 10, 11, 20, 12, 22, 30, 13, 23, 33, 40, 14, 24, 34, 44, 50, 15, 25, 35, 45, 55, 60, 16, 26, 36, 46, 56, 66, 70, 17, 27, 37, 47, 57, 67, 77, 80, 18, 28, 38, 48, 58, 68, 78, 88, 90, 19, 29, 39, 49, 59, 69, 79, 89, 99, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122

Offset: 0

Scott R. Shannon, Nov 21 2019

This is a variation on A329447. Instead of just considering the number of times the digits 0 to 9 have appeared so far in the sequence, we consider all numbers formed by the substrings of each previous entry; from single digits to the entire entry. Each value of a(n) is determined by considering the count of all the substrings occurring in a(0) to a(n-1) and then choosing the one which forms the lowest number when the count of that substring is concatenated with the substring itself.
The only restriction in the substring counting is that any substrings which have leading zeros are ignored. For example if a(n) was 2001, this would lead to incrementing the count of '0' by two, the count of '1' by one, the count of '2' by one, the count of '20' by 1, the count of '200' by 1, and the count of '2001' by 1. The substrings '00','01', and '001' are ignored and do not increment the count of '0' or '1'.
Unlike A329447, which is fairly constant in its growth, this sequence can have large decreases in its values from one entry to the next. These tend to be bunched and are followed by long series of entries with steady growth. For n up to 200000 the largest entry is 434397, the sequence decreases 4690 times, and the largest drop from one value to the next is 419882 which occurs at n = 199902.
This sequence is the same as A329447 up to a(55) = 100. After that, as a(2) = 11 and '11' has only appeared once, the next smallest value is that of 'one 11', that is 111.

			a(56) = 111 as a(2) = 11, and as 11 has only appeared once, the next smallest value is that of 'one 11', that is 111. After this entry the string '11' has now appeared three times, and the string '111' has appeared once.
a(199903) = 14342 as a(199902) = 434224 and as that contains the first appearance of the digit string '4342' in any entry, the next smallest value is that of 'one 4342', that is 14342. This leads to the largest drop in value for the first 200000 terms of the sequence. As a comparison after 199902 terms the digit '1' has occurred 153333 times.

Scott R. Shannon, Table of n, a(n) for n = 0..20000
Eric Angelini, Look left and say, Nov 14 2019.
Scott R. Shannon, Scatter plot for the first 200000 terms.

Cf. A005150, A045918, A329447.

A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554

Offset: 1

Amarnath Murthy, Aug 10 2005

Except for initial 0, rearrangement of numbers in A056524. They first differ at a(101) = 110011, while A056524(101) = 101101. If n has digits d_1 d_2 ... d_k, permute them to d_1 d_k d_2 d_{k-1} ... d_{floor(k/2)+1} and use that as index to A056524. - Franklin T. Adams-Watters, Jun 20 2006

			a(12) = 1221, deleting the LSB and the third digit 2 we get 12, deleting second and fourth digit we get 21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A045918.

Cf. A056524, A031298.

import Data.List (transpose)
a110745 n = read (concat $ transpose [ns, reverse ns]) :: Integer
            where ns = show n
-- Reinhard Zumkeller, Feb 14 2015

More terms from Franklin T. Adams-Watters, Jun 20 2006

A260526 Concatenate the positions of digits 0, 1,..., 9 in the decimal representation of n, using 1 for the rightmost digit etc., skip it if a digit does not occur.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 12, 21, 21, 21, 21, 21, 21, 21, 21, 12, 12, 12, 21, 21, 21, 21, 21, 21, 21, 12, 12, 12, 12, 21, 21, 21, 21, 21, 21, 12, 12, 12, 12, 12, 21, 21, 21, 21, 21, 12, 12, 12, 12, 12, 12, 21, 21, 21, 21, 12, 12, 12, 12, 12, 12, 12, 21, 21

Offset: 0

M. F. Hasler, Jul 28 2015

Sequence A260274 lists fixed points of this function.

Sequence A260519 is the same except for concatenating a 0 for digits which do not occur.

			Consider n=1103, the digit '0' is at 2nd position (looking from the right), digit '1' in 3rd and 4th place, digit '3' at 1st position, digits 2 and 4-9 do not occur. Thus a(1103) = 2341.

Cf. A260527 - A260529 and A260519 - A260522 for variants; A045918 for the classical "look and say" sequence.

a(n)={my(p=vector(10));for(i=1,#n=if(n,Vecrev(digits(n)),[0]),p[n[i]+1]=if(p[n[i]+1],concat(p[n[i]+1],i),i));if(1<#p=concat(select(t->#t,p)),p[1]=Str(p[1]);eval(concat(p)),p)}

A113589 A complementary variation of 'n described': if n is read as "a ones b twos" then a(n) = "one a's two b's", etc.

Original entry on oeis.org

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1, 2, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11, 111, 22, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111, 1111, 11111, 222

Offset: 0

Amarnath Murthy, Nov 07 2005

			A045918(10) = 1110 = 'one one one zero', hence a(10) = 'one one, zero ones' = 1.
A045918(11) = 21 = 'two ones', hence a(11) = 'one twos' = 2.
a(123) = 111111.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A002275, A045918, A140057, A306366.

a(n) = { my (d=[]); while (n, my (t=n%10, k=0); while (n%10==t, n\=10; k++;); for (i=1, t, d=concat(digits(k), d))); fromdigits(d) } \\ Rémy Sigrist, Feb 09 2022

a(A002275(n)) = n. - Rémy Sigrist, Feb 09 2022

Extended and corrected by Nathaniel Johnston, Apr 29 2011

a(0) = 0 prepended by Rémy Sigrist, Feb 09 2022

A321226 Describe the binary representation of n in binary and convert back to decimal.

Original entry on oeis.org

2, 3, 14, 5, 28, 59, 22, 7, 30, 115, 238, 117, 44, 91, 30, 9, 56, 123, 462, 229, 476, 955, 470, 119, 46, 179, 366, 181, 60, 123, 38, 11, 58, 227, 494, 245, 924, 1851, 918, 231, 478, 1907, 3822, 1909, 940, 1883, 478, 233, 88, 187, 718, 357, 732, 1467, 726, 183

Offset: 0

Rémy Sigrist, Nov 10 2018

This sequence is a binary variant of the "Look and Say" sequence A045918.

There is only one fixed point: a(7) = 7.

			For n = 67:
- the binary representation of 67 is "1000011",
- we see, in binary: "1" "1", "100" "0", "10" "1",
- hence the binary representation of a(67) is "111000101",
- and a(67) = 453 in decimal.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A020330, A045918, A049190.

a(n, b=2) = if (n==0, return (b)); my (d=digits(b*n, b), v=0, w=0); d[#d] = -1; for (i=1, #d-1, w++; if (d[i]!=d[i+1], v = b*(v*b^#digits(w, b) + w) + d[i]; w = 0)); v

a(2^n - 1) = 2*n + 1 for any n > 0.
a(4*n + 1) = 4*a(2*n) + 3 for any n > 0.
a(4*n + 2) = 4*a(2*n + 1) + 2 for any n >= 0.
a(A020330(2*n)) = A020330(a(2*n)) for any n > 0.
a(A049190(n)) = A049190(n+1) for any n > 0.

A321485 Describe n from left to right in terms of blocks with the maximum number of repetitions (and then with the maximum size).

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 21, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 22, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 23, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 24, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154

Offset: 0

Rémy Sigrist, Nov 11 2018

This sequence is a variant of the "Look and Say" sequence A045918 with features from Gijswijt's sequence A090822.

Any digit appearing in n also appears in a(n).

			The following table shows some examples with the corresponding blocks:
  n         a(n)    Blocks
  --------  ------  ---------------
         0      10              (0)
         1      11              (1)
        12     112             (12)
       100    1100            (100)
       111      31          (1|1|1)
     22123  221123       (2|2)(123)
    112211  212221  (1|1)(2|2)(1|1)
    212212    2212        (212|212)
  11221122   21122      (1122|1122)

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A321485

Cf. A045918, A090822.

PARI
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See Links section.
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a((10^k-1)/9 * d) = 10*k + d for any k > 0 and d = 1..9.

A055642(a(n)) <= 1 + A055642(n).

cp's OEIS Frontend

A049190 Start with 1. Convert to base 2, describe it in base 2, convert to base 10. Repeat with the new result.

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PARI

Extensions

A045981 Describe all the previous terms!.

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Haskell

Extensions

A260521 Concatenate the positions of digits 9, 8,..., 0 in the decimal representation of n, using 1 for the rightmost digit etc., and 0 when the digit does not occur.

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PARI

A260529 Concatenate the positions of digits 9, 8,..., 0 in the decimal representation of n, using 1 for the rightmost digit etc., skip it if a digit does not occur.

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Crossrefs

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PARI

A329818 Look left and tell the least frequent digit string; a(0) = 0, a(n) = concat(c,s), where c > 0 is the number of times the string s, composed of one or more digits, has appeared in a(0) to a(n-1) such that a(n) is minimal. Digit strings with leading zeros are ignored.

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A110745 a(n) is a number such that if odd positioned digits are deleted one gets n and if even positioned digits are deleted one gets n reversed. Counting is from the LSB side. The position of LSB is one.

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Haskell

Extensions

A260526 Concatenate the positions of digits 0, 1,..., 9 in the decimal representation of n, using 1 for the rightmost digit etc., skip it if a digit does not occur.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A113589 A complementary variation of 'n described': if n is read as "a ones b twos" then a(n) = "one a's two b's", etc.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A321226 Describe the binary representation of n in binary and convert back to decimal.

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