A045918 -id:A045918 - cp's OEIS frontend

A079342 Integers k that divide LS(k), where LS is the "Look and Say" function (A045918).

1, 2, 5, 10, 22, 32, 62, 91, 183, 188, 190, 196, 258, 276, 330, 671, 710, 1130, 1210, 1570, 2644, 2998, 3292, 4214, 17055, 20035, 53015, 70315, 101010, 108947, 199245, 233606, 309665, 323232, 356421, 483405, 626262, 919191, 1743599

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003

Infinite since s^i is a term for all odd i and s = 10, 32, 62, 91, 183, 190, 196, 258, 276, 671, 710, 1210, 1570, ..., where ^ denotes repeated concatenation of digits. - Michael S. Branicky, Aug 28 2024

			E.g. LS(1)=11, LS(2)=12, LS(10)=1110, LS(188)=1128 etc. and in each case LS(n) is a multiple of n.
122918=0 mod 2998, so 2998 is in the sequence.
But 13 == 1 mod 3, so 3 is not in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..82 (all terms <= 10^10)

Cf. A056815, A005150, A079562.

Cf. A152957. - David Wasserman, Dec 15 2008

# Implementation by R. J. Mathar, May 08 2019:
A045918 := proc(n)
    local a,f,pd,dgs,i ;
    a := [] ;
    f := 0 ;
    pd := -1 ;
    dgs := convert(n,base,10) ;
    for i from 1 to nops(dgs) do
        if op(-i,dgs) <> pd then
            if pd >= 0 then
                a := [op(a),f,pd] ;
            end if;
            pd := op(-i,dgs) ;
            f := 1 ;
        else
            f:= f+1 ;
        end if;
    end do:
    a := [op(a),f,pd] ;
    digcatL(%) ;
end proc:
isA079342 := proc(n)
    simplify( modp(A045918(n) ,n) = 0 ) ;
end proc:
for n from 1 to 30000 do
    if isA079342(n) then
        print(n) ;
    end if;
end do:

def LS(n): return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
def ok(n): return LS(n)%n == 0
print([k for k in range(1, 10**4) if ok(k)]) # Michael S. Branicky, Aug 28 2024

A369092 Numbers which are a substring of their own "Look and Say" description (cf. A045918).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 31, 41, 51, 61, 71, 81, 91, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 141, 151, 161, 171, 181, 191, 221, 322, 422, 522, 622, 722, 822, 922, 1022, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191

Offset: 1

Scott R. Shannon, Jan 13 2024

It is likely the 171 terms of the attached bfile, with a(171) = 22119221, forms the complete list of such numbers, although this is unknown.

			1 is a term as A045918(1) = 10, which contains "1" as a substring.
101 is a term as A045918(101) = 111011, which contains "101" as a substring.
22119221 is a term as A045918(22119221) = 2221192211, which contains "22119221" as a substring. It is likely this is the last possible term.

Scott R. Shannon, Table of n, a(n) for n = 1..171

Cf. A369132 (description is a substring of the number), A005150, A045918.

A369132 Numbers whose "Look and Say" description (cf. A045918) is a substring of the number.

Original entry on oeis.org

22, 333, 4444, 22333, 33322, 55555, 224444, 333000, 333111, 333222, 333444, 333555, 333666, 333777, 333888, 333999, 444422, 666666, 2255555, 3334444, 4444333, 5555522, 7777777, 22333000, 22333111, 22333222, 22333444, 22333555, 22333666, 22333777, 22333888, 22333999, 22666666, 33322333

Offset: 1

Scott R. Shannon, Jan 14 2024

This sequence is infinite as it contains A002275(A002275(k)) for any k > 1. - Rémy Sigrist, Jan 18 2024

			22 is a term as A045918(22) = 22, and "22" contains "22" as a substring.
333 is a term as A045918(333) = 33, and "333" contains "33" as a substring.
33322333 is a term as A045918(33322333) = 332233, and "33322333 " contains "332233" as a substring.
999999999 is a term as A045918(999999999) = 99, and "999999999" contains "99" as a substring. It is likely this is the last possible term.

Scott R. Shannon, Table of n, a(n) for n = 1..56

Cf. A002275, A005150, A045918, A369092 (number is a substring of its description).

A367974 Numbers which contain the "Look and Say" description (cf. A045918) of all their prime factors, counted with multiplicity.

Original entry on oeis.org

1, 25, 1024, 6272, 1953125, 4117715, 15813251, 213797679, 346146025, 488281250, 714592137, 1719341824, 3676531250, 10510100501, 10852734375, 11214315503, 17241013443, 25421511971

Offset: 1

Scott R. Shannon, Dec 07 2023

Overlapping of the "Look and Say" prime factor description strings is allowed. It is likely, although unproven, that 25 = 5*5 = "two 5's" = "25" is the only number that "perfectly" describes its own "Look and Say" factorization, i.e., there are no overlapping factor description strings, and all digits of the number are used in the factor description strings. It is unknown if the sequence is infinite.
There are many terms of the form 5^k, where k is 2, 9, 55, 62, 71, 82, 84, 86, 125, etc. - Ivan N. Ianakiev, Dec 07 2023
3262027661312 is a term. - Martin Ehrenstein, Dec 08 2023
10852734375 is a term. - Michael S. Branicky, Dec 08 2023
10510100501 is a term. - Michael S. Branicky, Dec 09 2023

			1 is a term since it has no prime factors.
25 is a term as 25 = 5*5, i.e., two 5's being "25", which appears in 25.
1024 is a term as 1024 = 2^10, i.e., ten 2's being "102", which appears in 1024.
346146025 is a term as 346146025 = 5^2 * 61^4, i.e., two 5's and four 61's being "25" and "461", respectively, both of which appear in 346146025.
1719341824 is a term as 1719341824 = 2^8 * 719 * 9341, i.e., eight 2's and one 719 and one 9341 being "82", "1719" and "19341" respectively, all of which appear in 1719341824. Note that only the final digit 4 is not used in the string descriptions.

Cf. A045918, A005150, A027746.

l[n_]:=ToString/@Reverse[Flatten[FactorInteger[n]]]; len[n_]:=Length[l[n]];
fQ[n_]:=AllTrue[Table[StringJoin[l[n][[i]],l[n][[i+1]]],{i,1,len[n],2}], StringPosition[ToString[n],#]!={}&]; Select[Range[6272],fQ[#]&] (* Ivan N. Ianakiev, Dec 07 2023 *)

from sympy import factorint
def ok(n):
    s = str(n)
    return all(str(e)+str(p) in s for p, e in factorint(n).items())
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 08 2023

1 prepended by Martin Ehrenstein, Dec 08 2023
a(14)-a(16) from Michael S. Branicky, Dec 13 2023
a(17)-a(18) from Michael S. Branicky, Dec 27 2023

A379453 Numbers that decrease when they are replaced by the "Look and Say" description (cf. A045918) of their prime factors, counted with multiplicity.

Original entry on oeis.org

49, 64, 81, 125, 128, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1250, 1331, 1369, 1458, 1681, 1715, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2500, 2809, 2916, 3087, 3125, 3375, 3481, 3721, 4096, 4374, 4489, 4802, 4913, 5000, 5041, 5329, 5488, 5625, 5832, 6075, 6125, 6241, 6250, 6561, 6859, 6889, 7203, 7776, 7921, 8000, 8192, 8575, 8748

Offset: 1

Scott R. Shannon, Dec 23 2024

			49 is a term as 49 = 7^2 which becomes 27 when replaced by the "Look and Say" description of its prime factors, and 27 is smaller than 49.

Scott R. Shannon, Table of n, a(n) for n = 1..5000

Cf. A045918, A379454, A379455, A005150, A367974, A369132, A369092.

Select[Range[8750],#>FromDigits[Flatten[IntegerDigits/@Reverse/@FactorInteger[#]]]&] (* James C. McMahon, Dec 23 2024 *)

A379455 Numbers that decrease two times in succession when they are iteratively replaced by the "Look and Say" description (cf. A045918) of their prime factors, counted with multiplicity.

Original entry on oeis.org

1849, 7921, 38809, 79507, 146529, 160801, 200978, 226981, 327697, 654481, 1113032, 1653125, 1731619, 1765376, 2109375, 2588881, 3418927, 3857868, 4182703, 5640625, 6492304, 6553600, 6892100, 7103125, 7845601, 8438707, 9509327, 11039981, 11880448, 12068352, 12106067, 12584111, 13227109, 14895500, 16843208, 17149469, 17372224, 18081075, 18852697, 19523584

Offset: 1

Scott R. Shannon, Dec 23 2024

nonn

			1849 is a term as 1849 = 43^2 which becomes 243 when replaced by the "Look and Say" description of its prime factors, and 243 is smaller than 1849, and 243 = 3^5 which becomes 53 when replaced by the "Look and Say" description of its prime factors, and 53 is smaller than 243.

Scott R. Shannon, Table of n, a(n) for n = 1..568

Cf. A045918, A379453, A379454, A005150, A367974, A369132, A369092.

A036978 Numbers for which the "describe what you see" transform A045918 produces a prime.

Original entry on oeis.org

1, 3, 7, 9, 17, 23, 27, 33, 39, 51, 63, 69, 81, 93, 99, 111, 113, 121, 127, 133, 137, 161, 193, 199, 203, 209, 223, 233, 271, 299, 301, 311, 313, 323, 331, 343, 359, 361, 367, 371, 373, 377, 431, 433, 439, 441, 447, 451, 463, 469, 477, 479, 481, 497

Offset: 1

G. L. Honaker, Jr.

			E.g. 33 -> "Two threes" -> 23, which is prime.

Cf. A036979, A045918.

for(n=1,999,isprime(A045918(n)) & print1(n","))  \\ M. F. Hasler, Jan 27 2012

Corrected by inserting the missing a(19)=127 by M. F. Hasler, Jan 27 2012

A097601 Differences between A045918 and A047842.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 198, 99, 0, 0, 0, 0, 0, 0, 0, 0, 297, 198, 99, 0, 0, 0, 0, 0, 0, 0, 396, 297, 198, 99, 0, 0, 0, 0, 0, 0, 495, 396, 297, 198, 99, 0, 0, 0, 0, 0, 594, 495, 396, 297, 198, 99, 0, 0, 0, 0, 693, 594, 495, 396, 297, 198, 99, 0, 0

Offset: 0

Odimar Fabeny, Aug 29 2004

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A045918, A047842.

LookAndSayA[n_]:= FromDigits@Flatten@IntegerDigits@Flatten[Through[ {Length, First}[#]] & /@ Split@IntegerDigits@n]; dc[n_]:= FromDigits@ Flatten@Select[Table[{DigitCount[n, 10, k], k}, {k, 0, 9}], #[[1]] > 0 &]; Table[LookAndSayA[n] - dc[n], {n, 0, 100}]

Terms a(31) onward added by G. C. Greubel, Jan 14 2019

A121993 Numbers k that yield a smaller number a(k) under the "Look and Say" function A045918.

Original entry on oeis.org

33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 1222, 1333, 1444, 1555, 1666, 1777, 1888, 1999, 2000, 2111, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 2333, 2444, 2555, 2666, 2777, 2888, 2999, 3000, 3111, 3222, 3300, 3311

Offset: 1

Sergio Pimentel, Sep 11 2006

			a(26)=2000 because under the Look and Say operator, 2000 is described as one two three zeros or: 1230, which is smaller than 2000.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Look and Say Sequence.

Cf. A045918, A005150, A005341, A023989, A079475.

a121993 n = a121993_list !! (n-1)
a121993_list = filter (\x -> a045918 x < x) [0..]
-- Reinhard Zumkeller, Jan 25 2014

from itertools import groupby
def ok(n): return n > int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
print([k for k in range(3312) if ok(k)]) # Michael S. Branicky, May 26 2023

A152957 Numbers n such that LS(n) divides n, where LS is the "Look and Say" function (A045918).

Original entry on oeis.org

22, 777, 4444, 200000, 333000, 333333, 555555, 660000, 666666, 700000, 4444400, 7700000, 22333000, 44445555, 55556666, 77700000, 88888888, 200000000, 777770000, 999900000, 999999000, 2222220000, 22222200000, 22333338888, 28800000000, 35555555505, 111111000000, 111777000000

Offset: 1

David Wasserman, Dec 15 2008

			777 is a member because it is three 7's, so LS(777) = 37 which divides 777.

Cf. A045918, A079342.

Definition corrected and more terms from Sean A. Irvine, Mar 17 2011

cp's OEIS Frontend

A079342 Integers k that divide LS(k), where LS is the "Look and Say" function (A045918).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A369092 Numbers which are a substring of their own "Look and Say" description (cf. A045918).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A369132 Numbers whose "Look and Say" description (cf. A045918) is a substring of the number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A367974 Numbers which contain the "Look and Say" description (cf. A045918) of all their prime factors, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Extensions

A379453 Numbers that decrease when they are replaced by the "Look and Say" description (cf. A045918) of their prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A379455 Numbers that decrease two times in succession when they are iteratively replaced by the "Look and Say" description (cf. A045918) of their prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

A036978 Numbers for which the "describe what you see" transform A045918 produces a prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A097601 Differences between A045918 and A047842.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A121993 Numbers k that yield a smaller number a(k) under the "Look and Say" function A045918.

Views

Author

Keywords

Examples

Links

Crossrefs