A043562 -id:A043562 - cp's OEIS frontend

A005150 Look and Say sequence: describe the previous term! (method A - initial term is 1).

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, 11131221133112132113212221, 3113112221232112111312211312113211, 1321132132111213122112311311222113111221131221, 11131221131211131231121113112221121321132132211331222113112211, 311311222113111231131112132112311321322112111312211312111322212311322113212221

Offset: 1

N. J. A. Sloane

Method A = "frequency" followed by "digit"-indication.
Also known as the "Say What You See" sequence.
Only the digits 1, 2 and 3 appear in any term. - Robert G. Wilson v, Jan 22 2004
All terms end with 1 (the seed) and, except the third a(3), begin with 1 or 3. - Jean-Christophe Hervé, May 07 2013
Proof that 333 never appears in any a(n): suppose it appears for the first time in a(n); because of "three 3" in 333, it would imply that 333 is also in a(n-1), which is a contradiction. - Jean-Christophe Hervé, May 09 2013
This sequence is called "suite de Conway" in French (see Wikipédia link). - Bernard Schott, Jan 10 2021
Contrary to many accounts (including an earlier comment on this page), Conway did not invent the sequence. The first mention of the sequence appears to date back to the 1977 International Mathematical Olympiad in Belgrade, Yugoslavia. See the Editor's note on page 4, directly preceding Conway's article in Eureka referenced below. - Harlan J. Brothers, May 03 2024

			The term after 1211 is obtained by saying "one 1, one 2, two 1's", which gives 111221.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 208.
S. R. Finch, Mathematical Constants, Cambridge, 2003, section 6.12 Conway's Constant, pp. 452-455.
M. Gilpin, On the generalized Gleichniszahlen-Reihe sequence, Manuscript, Jul 05 1994.
A. Lakhtakia and C. Pickover, Observations on the Gleichniszahlen-Reihe: An Unusual Number Theory Sequence, J. Recreational Math., 25 (No. 3, 1993), 192-198.
Clifford A. Pickover, Computers and the Imagination, St Martin's Press, NY, 1991.
Clifford A. Pickover, Fractal horizons: the future use of fractals, New York: St. Martin's Press, 1996. ISBN 0312125992. Chapter 7 has an extensive description of the elements and their properties.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 486.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, 1999, p. 23.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.

T. D. Noe, Table of n, a(n) for n = 1..25
Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, Stuttering Conway Sequences Are Still Conway Sequences, arXiv:2006.06837 [math.DS], 2020.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, The Look-and-Say The Biggest Sequence Eventually Cycles, arXiv:2006.07246 [math.DS], 2020.
Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.
Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
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.
J. H. Conway and Brady Haran, Look-and-Say Numbers (2014), Numberphile video.
S. B. Ekhad and D. Zeilberger, Proof of Conway's Lost Cosmological Theorem, arXiv:math/9808077 [math.CO], 1998.
S. B. Ekhad and D. Zeilberger, Proof of Conway's lost cosmological theorem, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 78-82.
S. Eliahou and M. J. Erickson, Mutually describing multisets and integer partitions, Discrete Mathematics, Volume 313, Issue 4, Feb 28 2013, Pages 422-433. - From _N. J. A. Sloane_, Jan 03 2013
S. R. Finch, Conway's Constant [From the Wayback Machine]
Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163. See p. 161.
M. Hilgemeier, Die Gleichniszahlen-Reihe, in Bild der Wissenschaft, 12 (1986), 194-195, with permission from the Konradin Medien GmbH.
M. Hilgemeier, One metaphor fits all, in Fractal Horizons, ed. C. A Pickover, St. Martins, NY, 1996, pp. 137-161.
R. A. Litherland, Conway's Cosmological Theorem (Overview).
R. A. Litherland, Conway's Cosmological Theorem, 12 pages, Apr 14 2006 (pdf file).
R. A. Litherland, Programs for Conway's Cosmological Theorem, (gzipped tar ball).
R. A. Litherland, The audioactive package.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 37, etc.
MacTutor History of Mathematics, John H. Conway
O. Martin, Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA, Amer. Math. Monthly, 113 (No. 4, 2006), 289-307. - From _N. J. A. Sloane_, Feb 19 2013
Thomas Morrill, Look, Knave, arXiv:2004.06414 [math.CO], 2020.
Paulo Ortolan, Java program for A005150.
Matt Parker, Can you trust an elegant conjecture?, Stand-Up Maths, 2022, video.
Rosetta Code, Look and say sequence programs in over 60 languages.
J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.
T. Sillke, Conway sequence.
L. J. Upton, Letter to N. J. A. Sloane, Jan 08 1991.
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.
Wikipédia, Suite de Conway.
W. W. Zadrozny, Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence, arXiv:2109.12755 [cs.AI], 2022.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, A blind spot for large language models: Supradiegetic linguistic information, arXiv:2306.06794 [cs.CL], 2023.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Alejandro J. Ruiz, Calla Beauregard, Ashley Fehr, Mikaela Irene Fudolig, Bradford Demarest, Yoshi Meke Bird, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, Tokens, the oft-overlooked appetizer: Large language models, the distributional hypothesis, and meaning, arXiv:2412.10924 [cs.CL], 2024. See pp. 21, 28.

Cf. A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154, A007651, A060857.
Cf. A001387, Periodic table: A119566.
Cf. A225224, A221646, A225212 (continuous versions).
Apart from the first term, all terms are in A001637.
About digits: A005341 (number of digits), A022466 (number of 1's), A022467 (number of 2's), A022468 (number of 3's), A004977 (sum of digits), A253677 (product of digits).
About primes: A079562 (number of distinct prime factors), A100108 (terms that are primes), A334132 (smallest prime factor).
Cf. A014715 (Conway's constant), A098097 (terms interpreted as written in base 4).

import List
say :: Integer -> Integer
say = read . concatMap saygroup . group . show
where saygroup s = (show $ length s) ++ [head s]
look_and_say :: [Integer]
look_and_say = 1 : map say look_and_say
-- Josh Triplett (josh(AT)freedesktop.org), Jan 03 2007

a005150 = foldl1 (\v d -> 10 * v + d) . map toInteger . a034002_row
-- Reinhard Zumkeller, Aug 09 2012

Java
```
See Paulo Ortolan link.
```

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[ FromDigits[ F[ n ] ], {n, 1, 15} ]
A005150[1] := 1; A005150[n_] := A005150[n] = FromDigits[Flatten[{Length[#], First[#]}&/@Split[IntegerDigits[A005150[n-1]]]]]; Map[A005150, Range[25]] (* Peter J. C. Moses, Mar 21 2013 *)

A005150(n,a=1)={ while(n--, my(c=1); for(j=2,#a=Vec(Str(a)), if( a[j-1]==a[j], a[j-1]=""; c++, a[j-1]=Str(c,a[j-1]); c=1)); a[#a]=Str(c,a[#a]); a=concat(a)); a }  \\ M. F. Hasler, Jun 30 2011

$str="1"; for (1 .. shift(@ARGV)) { print($str, ","); @a = split(//,$str); $str=""; $nd=shift(@a); while (defined($nd)) { $d=$nd; $cnt=0; while (defined($nd) && ($nd eq $d)) { $cnt++; $nd = shift(@a); } $str .= $cnt.$d; } } print($str);
# Jeff Quilici (jeff(AT)quilici.com), Aug 12 2003

# This outputs the first n elements of the sequence, where n is given on the command line.
$s = 1;
for (2..shift @ARGV) {
print "$s, ";
$s =~ s/(.)\1*/(length $&).$1/eg;
}
# Arne 'Timwi' Heizmann (timwi(AT)gmx.net), Mar 12 2008
print "$s\n";

def A005150(n):
    p = "1"
    seq = [1]
    while (n > 1):
        q = ''
        idx = 0 # Index
        l = len(p) # Length
        while idx < l:
            start = idx
            idx = idx + 1
            while idx < l and p[idx] == p[start]:
                idx = idx + 1
            q = q + str(idx-start) + p[start]
        n, p = n - 1, q
        seq.append(int(p))
    return seq
# Olivier Mengue (dolmen(AT)users.sourceforge.net), Jul 01 2005

def A005150(n):
    seq = [1] + [None] * (n - 1) # allocate entire array space
    def say(s):
        acc = '' # initialize accumulator
        while len(s) > 0:
            i = 0
            c = s[0] # char of first run
            while (i < len(s) and s[i] == c): # scan first digit run
                i += 1
            acc += str(i) + c # append description of first run
            if i == len(s):
                break # done
            else:
                s = s[i:] # trim leading run of digits
        return acc
    for i in range(1, n):
        seq[i] = int(say(str(seq[i-1])))
    return seq
# E. Johnson (ejohnso9(AT)earthlink.net), Mar 31 2008

# program without string operations
def sign(n): return int(n > 0)
def say(a):
    r = 0
    p = 0
    while a > 0:
        c = 3 - sign((a % 100) % 11) - sign((a % 1000) % 111)
        r += (10 * c + (a % 10)) * 10**(2*p)
        a //= 10**c
        p += 1
    return r
a = 1
for i in range(1, 26):
    print(i, a)
    a = say(a)
# Volker Diels-Grabsch, Aug 18 2013

import re
def lookandsay(limit, sequence = 1):
    if limit > 1:
        return lookandsay(limit-1, "".join([str(len(match.group()))+match.group()[0] for matchNum, match in enumerate(re.finditer(r"(\w)\1*", str(sequence)))]))
    else:
        return sequence
# lookandsay(3) --> 21
# Nicola Vanoni, Nov 29 2016

import itertools
x = "1"
for i in range(20):
    print(x)
    x = ''.join(str(len(list(g)))+k for k,g in itertools.groupby(x))
# Matthew Cotton, Nov 12 2019

a(n+1) = A045918(a(n)). - Reinhard Zumkeller, Aug 09 2012
a(n) = Sum_{k=1..A005341(n)} A034002(n,k)*10^(A005341(n)-k). - Reinhard Zumkeller, Dec 15 2012
a(n) = A004086(A007651(n)). - Bernard Schott, Jan 08 2021
A055642(a(n+1)) = A005341(n+1) = 2*A043562(a(n)). - Ya-Ping Lu, Jan 28 2025
Conjecture: DC(a(n)) ~ k * (Conway's constant)^n where k is approximately 1.021... and DC denotes the number of digit changes in the decimal representation of n (e.g., DC(13112221)=4 because 1->3, 3-1, 1->2, 2->1). - Bill McEachen, May 09 2025
Conjecture: lim_{n->infinity} (c2+c3-c1)/(c1+c2+c3) = 0.01 approximately, where ci is the number of appearances of 'i' in a(n). - Ya-Ping Lu, Jun 05 2025

A297770 Number of distinct runs in base-2 digits of n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 3, 3, 2, 3, 4, 3, 3, 3, 2, 3, 4, 3, 3, 3, 2, 3, 4, 2, 4, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 1, 2, 2, 3, 3, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 4, 3, 3, 3, 3, 4, 3, 2, 3

Offset: 1

Clark Kimberling, Jan 26 2018

Every positive integers occurs infinitely many times.
***
Guide to related sequences:
Base b      # runs    # distinct runs
      A005811       A297770
      A043555       A297771
      A043556       A297772
      A043557       A297773
      A043558       A297774
      A043559       A297775
      A043560       A297776
      A043561       A297777
      A043562       A297778
      A043563       A297779
      A043564       A297780
      A043565       A297781
      A043566       A297782
      A043567       A297783
      A043568       A297784

			27 in base-2: 1,1,0,1,1; three runs, of which 2 are distinct:  0 and 11, so that a(27) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A005811 (number of runs, not necessarily distinct).

b = 2; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]]
Table[s[n], {n, 1, 200}]

apply( {A297770(n)=my(r=[0,0], c); while(n, my(d=bitand(n,1), L=valuation(n+d, 2)); !bittest(r[1+d], L) && c++ && r[1+d] += 1<>=L); c}, [0..99]) \\ M. F. Hasler, Jul 13 2024

a(n) = my(s=strjoin(binary(n)), v=vecsort(concat(strsplit(s, "1"), strsplit(s, "0")), , 8)); #v-(v[1]==""); \\ Ruud H.G. van Tol, Aug 05 2024

from itertools import groupby
def A297770(n): return len(set(map(lambda x:tuple(x[1]),groupby(bin(n)[2:])))) # Chai Wah Wu, Jul 13 2024

A047726 Number of different numbers that are formed by permuting digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 6

Offset: 1

N. J. A. Sloane

The minimum value of a(A171102(n)) is 10*9!. - Altug Alkan, Jul 08 2016

			From 102 we get 102, 120, 210, 201, 12 and 21, so a(102)=6.
From 33950 with 5 digits, one '0', two '3', one '5' and one '9', we get 5! / (1! * 2! * 1! * 1!) = 60 different numbers and a(33950) = 60.  - _Bernard Schott_, Oct 20 2019

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.

Cf. A055098. Identical to A043537 and A043562 for n<100.

Cf. A179239. - Aaron Dunigan AtLee, Jul 14 2010

import Data.List (permutations, nub)
a047726 n = length $ nub $ permutations $ show n
-- Reinhard Zumkeller, Jul 26 2011

f:= proc(n) local L;
  L:= convert(n,base,10);
  nops(L)!/mul(numboccur(i,L)!,i=0..9);
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 08 2016

pd[n_]:=Module[{p=Permutations[IntegerDigits[n]]},Length[Union [FromDigits/@p]]]; pd/@Range[120]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=n=eval(Vec(Str(n)));(#n)!/prod(i=0,9,sum(j=1,#n,n[j]==i)!) \\ Charles R Greathouse IV, Sep 29 2011

A047726(n)={local(c=Vec(0,10)); apply(d->c[d+1]++, digits(n)); logint(n*10,10)!/prod(i=1,10,c[i]!)} \\ M. F. Hasler, Oct 18 2019

a(n) << n / (log_10 n)^4.5 by Stirling's approximation. - Charles R Greathouse IV, Sep 29 2011

a(n) = A000142(A055642(n))/Product_{k=0..9} A000142(A100910(n,k)). - Robert Israel, Jul 08 2016

Corrected by Henry Bottomley, Apr 19 2000

A137214 a(n) is the number of distinct decimal digits in 2^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 3, 5, 4, 4, 7, 6, 5, 4, 4, 4, 6, 6, 6, 9, 7, 7, 5, 6, 6, 7, 7, 8, 7, 7, 7, 6, 8, 7, 9, 8, 7, 8, 9, 7, 8, 9, 8, 7, 7, 8, 8, 7, 9, 8, 9, 9, 9, 9, 9, 9, 8, 9, 10, 9, 10, 7, 9, 8, 9, 9, 9, 8, 9, 10, 9, 9, 10, 9, 10, 9, 9, 10, 10, 10, 9, 8, 9, 9, 10, 10, 10, 10, 10

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Appears to be all 10's starting at a(169). - T. D. Noe, Apr 01 2014

			a(16) = 3 because 2^16 = 65536, which contains 3 distinct decimal digits [3,5,6].

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A043562, A043536.

A043537 := proc(n) nops(convert(convert(n,base,10),set)) ; end: A137214 := proc(n) A043537(2^n) ; end: seq(A137214(n),n=0..120) ; # R. J. Mathar, Mar 16 2008
a:=proc(n) options operator, arrow: nops(convert(convert(2^n,base,10),set)) end proc: seq(a(n),n=0..80); # Emeric Deutsch, Apr 02 2008

Table[Length[Union[IntegerDigits[2^n]]], {n, 0, 100}] (* T. D. Noe, Apr 01 2014 *)

def a(n): return len(set(str(2**n)))
print([a(n) for n in range(99)]) # Michael S. Branicky, Jul 23 2021

a(n) = A043537(2^n). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar and Emeric Deutsch, Mar 16 2008

A364121 Stolarsky representation of n.

Original entry on oeis.org

0, 1, 11, 10, 111, 101, 110, 1111, 100, 1011, 1101, 1110, 11111, 1010, 1001, 10111, 1100, 11011, 11101, 11110, 111111, 1000, 10101, 10011, 10110, 101111, 11010, 11001, 110111, 11100, 111011, 111101, 111110, 1111111, 10100, 10001, 101011, 10010, 100111, 101101

Offset: 1

Amiram Eldar, Jul 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Casey Mongoven, Description of Stolarsky Representations.

Cf. A001622, A007064, A200648, A200649, A200650, A200651, A200714.

Cf. A007088, A043562, A055641, A055642, A268643.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
a[n_] := FromDigits[stol[n]]; Array[a, 100]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
a(n) = fromdigits(stol(n));

Description of an algorithm for calculating a(n):
Let s(1) = {} be the empty set, and for n > 1, let s(n) be the sequence of digits of a(n). s(n) can be calculated recursively by:
1. If n = round(round(n/phi)*phi) then s(n) = s(floor(n/phi^2) + 1) U {0}, where phi is the golden ratio (A001622) and U denotes concatenation.
2. If n != round(round(n/phi)*phi) then s(n) = s(round(n/phi)) U {1}.
a(n) = A007088(A200714(n)).
A268643(a(n)) = A200649(n).
A055641(a(n)) = A200650(n).
A055642(a(n)) = A200648(n).
A043562(a(n)) = A200651(n)

A297778 Number of distinct runs in base-10 digits of n.

Original entry on oeis.org

Offset: 1

Clark Kimberling, Feb 03 2018

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. A043562(n) = a(n) for n=1..100, but not for n=101.

			14! in base-10: 2,7,0,0,1,7,3,0,1,3,0,0; ten runs, of which 6 are distinct, so that a(14!) = 6.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A043562 (number of runs, not necessarily distinct), A297770.

b = 10; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]]
Table[s[n], {n, 1, 200}]

A379929 Numbers that have the same number of prime factors, counted with multiplicity, as there are runs in their base-10 representation.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 14, 15, 21, 25, 26, 34, 35, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 115, 118, 119, 122, 124, 125, 130, 133, 138, 147, 148, 153, 154, 155, 164, 165, 166, 170, 171, 172, 174, 175, 177, 182, 186, 190, 195, 207, 212, 221, 226, 230, 231

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A001222(k) = A043562(k).

			a(5) = 10 is a term because 10 has two runs (1 and 0) and two prime factors, 2 and 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A043562, A379930, A379931.

filter:= proc(n) local L; L:= convert(n,base,10); nops(L) - numboccur(0, L[2..-1]-L[1..-2]) = numtheory:-bigomega(n) end proc:
select(filter, [$1..1000]);

A379929Q[n_] := PrimeOmega[n] == Length[Split[IntegerDigits[n]]];
Select[Range[300], A379929Q] (* Paolo Xausa, Jan 08 2025 *)

from sympy import primeomega
def ok(n): return primeomega(n) == len(s:=str(n)) - sum(1 for i in range(1, len(s)) if s[i-1] == s[i])
print([k for k in range(1, 232) if ok(k)]) # Michael S. Branicky, Jan 08 2025

A379930 Numbers that have the same number of divisors as there are runs in their base-10 representation.

Original entry on oeis.org

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 121, 169, 199, 211, 223, 227, 229, 233, 277, 289, 311, 331, 337, 361, 433, 443, 449, 499, 529, 557, 577, 599, 661, 677, 733, 773, 811, 841, 877, 881, 883, 887, 911, 961, 977, 991, 997, 1018, 1027, 1037, 1041

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A000005(k) = A043562(k).

			a(5) = 23 is a term because it has two runs, 2 and 3, and two divisors, 1 and 23.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A043562, A379929, A379931.

filter:= proc(n) local L; L:= convert(n, base, 10); nops(L) - numboccur(0, L[2..-1]-L[1..-2]) = numtheory:-tau(n) end proc:
select(filter, [$1..1000]);

A379930Q[n_] := DivisorSigma[0, n] == Length[Split[IntegerDigits[n]]];
Select[Range[1000], A379930Q] (* Paolo Xausa, Jan 08 2025 *)

A379931 Numbers whose maximum exponent in their prime factorization is the number of runs in their base-10 representation.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 12, 18, 20, 22, 25, 28, 33, 36, 45, 49, 50, 52, 55, 60, 63, 66, 68, 75, 76, 77, 84, 90, 92, 98, 100, 104, 108, 111, 116, 117, 120, 125, 135, 136, 152, 168, 184, 188, 189, 216, 220, 222, 225, 228, 232, 244, 248, 250, 264, 270, 280, 296, 297, 300, 312, 328, 332, 338, 343, 351

Offset: 1

Robert Israel, Jan 06 2025

Numbers k such that A051903(k) = A043562(k).

If k has r runs, maximum exponent m <= r, and is coprime to 10, then 10^(r+1) * k is a term. Therefore this sequence is infinite.

			a(10) = 22 is a term because 22 = 2 * 11 has maximum exponent 1, and one run in its base 10 representation.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A043562, A051903, A379929, A379930.

filter:= proc(n) local L; L:= convert(n, base, 10); nops(L) - numboccur(0, L[2..-1]-L[1..-2]) = max(ifactors(n)[2][..,2]) end proc:
select(filter, [$1..1000]);

A379931Q[n_] := n > 1 && Max[FactorInteger[n][[All, 2]]] == Length[Split[IntegerDigits[n]]];
Select[Range[400], A379931Q] (* Paolo Xausa, Jan 08 2025 *)

A360391 a(n) is the number of distinct sums of nonempty subsets of the digits of n.

Original entry on oeis.org

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

Offset: 0

Ctibor O. Zizka, Feb 06 2023

1 <= a(n) <= 2^(1 + floor(log_10(n))) - 1.

			k = 10: sums of digits are {0, 1, 0 + 1}, distinct sums of digits are {0, 1}, thus a(10) = 2.
k = 11: sums of digits are {1, 1 + 1}, distinct sums of digits are {1, 2}, thus a(11) = 2.
k = 12: sums of digits are {1, 2, 1 + 2}, distinct sums of digits are {1, 2, 3}, thus a(12) = 3.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A007953, A043562, A054054, A054055, A055642.

a[n_] := Length[Union[Total /@ Select[Subsets[IntegerDigits[n]], # != {} &]]]; Array[a, 100, 0] (* Amiram Eldar, Feb 06 2023 *)

from itertools import combinations as C
def a(n):
    v = list(map(int, str(n)))
    return len(set(sum(c) for r in range(1, len(v)+1) for c in C(v, r)))
print([a(n) for n in range(89)]) # Michael S. Branicky, Feb 19 2023

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