A297770 -id:A297770 - cp's OEIS frontend

A351204 Number of integer partitions of n such that every permutation has all distinct runs.

1, 1, 2, 3, 4, 5, 8, 9, 11, 14, 18, 20, 25, 28, 34, 41, 47, 53, 64, 72, 84, 98, 113, 128, 148, 169, 194, 223, 255, 289, 333, 377, 428, 488, 554, 629, 715, 807, 913, 1033, 1166, 1313, 1483, 1667, 1875, 2111, 2369, 2655, 2977, 3332, 3729, 4170, 4657, 5195, 5797, 6459

Offset: 0

Gus Wiseman, Feb 15 2022

nonn

Partitions enumerated by this sequence include those in which all parts are either the same or distinct as well as partitions with an even number of parts all of which except one are the same. - Andrew Howroyd, Feb 15 2022

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (3111)    (4111)     (521)
                                     (111111)  (211111)   (2222)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A000005.
The version for normal multisets is 2^(n-1) - A283353(n-3).
The complement is counted by A351203, ranked by A351201.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A144300, A329746, A351291.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!UnsameQ@@Split[#]&]=={}&]],{n,0,15}]

\\ here Q(n) is A000009.
Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ Andrew Howroyd, Feb 15 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 15 2022

A350952 The smallest number whose binary expansion has exactly n distinct runs.

Original entry on oeis.org

0, 1, 2, 11, 38, 311, 2254, 36079, 549790, 17593311, 549687102, 35179974591, 2225029922430, 284803830071167, 36240869367020798, 9277662557957324543, 2368116566113212692990, 1212475681849964898811391, 619877748107024946567312382, 634754814061593545284927880191

Offset: 0

Gus Wiseman, Feb 14 2022

nonn

Positions of first appearances in A297770 (with offset 0).

The binary expansion of terms for n > 0 starts with 1, then floor(n/2) 0's, then alternates runs of increasing numbers of 1's, and decreasing numbers of 0's; see Python code. Thus, for n even, terms have n*(n/2+1)/2 binary digits, and for n odd, ((n+1) + (n-1)*((n-1)/2+1))/2 binary digits. - Michael S. Branicky, Feb 14 2022

			The terms and their binary expansions begin:
       0:                   ()
       1:                    1
       2:                   10
      11:                 1011
      38:               100110
     311:            100110111
    2254:         100011001110
   36079:     1000110011101111
  549790: 10000110001110011110
For example, 311 has binary expansion 100110111 with 5 distinct runs: 1, 00, 11, 0, 111.

Michael S. Branicky, Table of n, a(n) for n = 0..114
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Runs in binary expansion are counted by A005811, distinct A297770.
The version for run-lengths instead of runs is A165933, for A165413.
Subset of A175413 (binary expansion has distinct runs), for lengths A044813.
The version for standard compositions is A351015.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A334028 counts distinct parts in standard compositions.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A070939, A085207, A098859, A233564, A325545, A328592, A333489, A351203, A351291.

q=Table[Length[Union[Split[If[n==0,{},IntegerDigits[n,2]]]]],{n,0,1000}];Table[Position[q,i][[1,1]]-1,{i,Union[q]}]

a(n)={my(t=0); for(k=1, (n+1)\2, t=((t<Andrew Howroyd, Feb 15 2022

def a(n): # returns term by construction
    if n == 0: return 0
    q, r = divmod(n, 2)
    if r == 0:
        s = "".join("1"*i + "0"*(q-i+1) for i in range(1, q+1))
        assert len(s) == n*(n//2+1)//2
    else:
        s = "1" + "".join("0"*(q-i+2) + "1"*i for i in range(2, q+2))
        assert len(s) == ((n+1) + (n-1)*((n-1)//2+1))//2
    return int(s, 2)
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 14 2022

a(9)-a(19) from Michael S. Branicky, Feb 14 2022

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset: 0

Gus Wiseman, Feb 12 2022

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

a(26) onwards from Andrew Howroyd, Jan 27 2024

A043562 Number of runs in base-10 representation of n.

Original entry on oeis.org

1, 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

Offset: 0

Clark Kimberling

Blecksmith, Filaseta, & Nicol show that lim a(k^n) = infinity whenever k is not a power of 10. More generally, in base b, the limit is infinity exactly when log k/log b is irrational. - Charles R Greathouse IV, Jan 29 2014

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. - Clark Kimberling, Feb 04 2018

Clark Kimberling, Table of n, a(n) for n = 0..10000
Richard Blecksmith, Michael Filaseta, and Charles Nicol, A result on the digits of a^n, Acta Arithmetica 64 (1993), pp. 331-339.

Cf. A297778 (number of distinct runs), A297770.

Table[Length[Split[IntegerDigits[n]]],{n,0,90}] (* Harvey P. Dale, Aug 24 2016 *)

a(n)=my(d=digits(n)); #d-sum(i=2,#d,d[i]==d[i-1]) \\ Charles R Greathouse IV, Jan 29 2014

def a(n): return len(s:=str(n))-sum(1 for i in range(1, len(s)) if s[i-1] == s[i])
print([a(n) for n in range(90)]) # Michael S. Branicky, Jan 08 2025 after Charles R Greathouse IV

A043555 Number of runs in base-3 representation of n.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Every positive integer occurs infinitely many times.  See A297770 for a guide to related sequences.
Having a(0) = 1 (rather than a(0) = 0) is debatable, on the grounds that a(0) = 1 is determined by our culture, rather than the underlying mathematics. See my August 2020 comment in A145204. - Peter Munn, Jul 12 2024
From M. F. Hasler, Jul 13 2024: (Start)
The base-2 version has a(0) = 0, corresponding to the convention that 0 has zero digits, which is the more logical (but maybe less human) convention, such that, e.g., b^n is the least number with n+1 digits in base b (<=> b^n - 1 is the largest number with n digits), valid also for 0. Here and in A043556 (base 4) the convention is that 0 has one digit, '0'.
"Runs" means substrings of consecutive equal digits, here in the base-3 representation of the numbers. See Example for details. (End)

			From _M. F. Hasler_, Jul 13 2024: (Start)
Numbers n = 0, 1, 2, 3, 4, 5, ... are written '0', '1', '2', '10', '11', '12', ... in base 3. The first three have one single digit, so there is just 1 "run" (= subsequence of equal digits), whence a(0) = a(1) = a(2) = 1.
In '10' we have a "run" of '1's of length 1, followed by a run of '0's of length 1, so there are a(3) = 2 runs.
In '11' we have again one single run, here of 2 digits '1', whence a(4) = 1. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..19682 (first 1001 terms from Zak Seidov)

Cf. A005811 (in base 2), A043556 (in base 4), A145204, A297770, A297771 (number of distinct runs).

Cf. A033113.

NRUNS := proc(L::list)
    local a,i;
    a := 1 ;
    for i from 2 to nops(L) do
        if op(i,L) <> op(i-1,L) then
            a := a+1 ;
        end if
    end do:
    a ;
end proc:
A043555 := proc(n)
    convert(n,base,3) ;
    NRUNS(%) ;
end proc:
seq(A043555(n),n=0..80) ; # R. J. Mathar, Jul 12 2024
# second Maple program:
a:= n-> `if`(n<3, 1, a(iquo(n, 3))+`if`(n mod 9 in {0, 4, 8}, 0, 1)):
seq(a(n), n=0..89);  # Alois P. Heinz, Jul 13 2024

b = 3; s[n_] := Length[Split[IntegerDigits[n, b]]];
Table[s[n], {n, 1, 200}]

a(n)=my(d=digits(n,3)); sum(i=2,#d,d[i]!=d[i-1])+1 \\ Charles R Greathouse IV, Jul 20 2014

from itertools import groupby
from sympy.ntheory import digits
def A043555(n): return len(list(groupby(digits(n,3)[1:]))) # Chai Wah Wu, Jul 13 2024

Updated by Clark Kimberling, Feb 03 2018

A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 26 2022

Assuming the binary digits are not all 1, this is one more than the number of different lengths of runs of 1's in the binary expansion of n.

			The binary expansion of 183 is (1,0,1,1,0,1,1,1), with runs (1), (0), (1,1), (0), (1,1,1), with sums 1, 0, 2, 0, 3, of which four are distinct, so a(183) = 4.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

For lengths of all runs we have A165413, firsts A165933.
Numbers whose binary expansion has distinct runs are A175413.
For runs instead of run-sums we have A297770, firsts A350952.
For prime indices we have A353835, weak A353861, firsts A006939.
For standard compositions we have A353849, firsts A246534.
Positions of first appearances are A353930.
A005811 counts runs in binary expansion.
A044813 lists numbers with distinct run-lengths in binary expansion.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Cf. A215203, A353743, A353832, A353847, A353932, A354579.

Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,0,100}]

from itertools import groupby
def A353929(n): return len(set(sum(map(int,y[1])) for y in groupby(bin(n)[2:]))) # Chai Wah Wu, Jun 26 2022

A353930 Smallest number whose binary expansion has n distinct run-sums.

Original entry on oeis.org

1, 2, 11, 183, 5871, 375775, 48099263, 12313411455, 6304466665215, 6455773865180671, 13221424875890015231, 54154956291645502388223, 443637401941159955564326911, 7268555193403964711965932118015, 238176016577461115681699663643131903, 15609103422420491677315869156516292427775

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The terms, binary expansions, and standard compositions begin:
       1:                    1  (1)
       2:                   10  (2)
      11:                 1011  (2,1,1)
     183:             10110111  (2,1,2,1,1,1)
    5871:        1011011101111  (2,1,2,1,1,2,1,1,1,1)
  375775:  1011011101111011111  (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Essentially the same as A215203.
For prime indices instead of binary expansion we have A006939.
For lengths instead of sums of runs we have A165933 = firsts in A165413.
Numbers whose binary expansion has all distinct runs are A175413.
For standard compositions we have A246534, firsts of A353849.
For runs instead of run-sums we have A350952, firsts of A297770.
These are the positions of first appearances in A353929.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
A353835 counts partitions with all distinct run-sums, weak A353861.
A353864 counts rucksack partitions.
Cf. A044813, A073093, A181819, A304442, A353743, A353840, A353841, A353842, A353847, A353848, A353850, A353853, A353932, A354582.

qe=Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,1,10000}];
Table[Position[qe,i][[1,1]],{i,Max@@qe}]

a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<Andrew Howroyd, Jan 01 2023

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023

A165933 Least integer, k, whose value is n in A165413.

Original entry on oeis.org

1, 4, 35, 536, 16775, 1060976, 135007759, 34460631520, 17617985239071, 18027600169142208, 36907002795598798911, 151143401509104346210176, 1238053384151947477501575295, 20283338091738780737237428502272, 664629209970464486086782992577855743

Offset: 1

Robert G. Wilson v, Sep 30 2009

An alternative name: The smallest number whose binary expansion has exactly n distinct run-lengths. - Gus Wiseman, Feb 21 2022

Term a(n) has one 1, followed by n 0's, then two 1's, (n-1) 0's, ..., up to n runs; see Python program. - Michael S. Branicky, Feb 22 2022

			a(1) in binary is 1, a(2) in binary is 100, a(3) in binary is 100011, a(4) in binary is 1000011000, etc.
From _Gus Wiseman_, Feb 21 2022: (Start)
The terms and their binary expansions begin:
  n              a(n)
  1:               1 =                                             1
  2:               4 =                                           100
  3:              35 =                                        100011
  4:             536 =                                    1000011000
  5:           16775 =                               100000110000111
  6:         1060976 =                         100000011000001110000
  7:       135007759 =                  1000000011000000111000001111
  8:     34460631520 =          100000000110000000111000000111100000
  9:  17617985239071 = 100000000011000000001110000000111100000011111
(End)

Michael S. Branicky, Table of n, a(n) for n = 1..81

A subset of A044813 (distinct run-lengths) and of A175413 (distinct runs).
These are the positions of first appearances in A165413.
The version for runs instead of run-lengths is A350952, firsts of A297770.
A000120 counts binary weight.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A329739 = compositions, for runs A351013.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A003242, A070939, A085207, A098859, A325545, A328592, A351015, A351202, A351203, A351291.

g[n_] := Table[ {Table[1, {i}], Table[0, {n - i + 1}]}, {i, Floor[(n + If[ OddQ@n, 1, 0])/2]}]; f[n_] := FromDigits[ If[ OddQ@n, Flatten@ Most@ Flatten[ g@n, 1], Flatten@ g@n], 2]; Array[f, 14]
s=Table[Length[Union[Length/@Split[IntegerDigits[n,2]]]],{n,0,1000}]; Table[Position[s,k][[1,1]]-1,{k,Union[s]}] (* Gus Wiseman, Feb 21 2022 *)

def a(n): # returns term by construction
    if n == 1: return 1
    q, r = divmod(n+1, 2)
    s = "".join("1"*i + "0"*(n+1-i) for i in range(1, q+1))
    if r == 0: s = s.rstrip("0")
    return int(s, 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 22 2022

a(15) and beyond from Michael S. Branicky, Feb 22 2022

A043556 Number of runs in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences.

Cf. A297772 (number of distinct runs), A297770.

b = 4; s[n_] := Length[Split[IntegerDigits[n, b]]];
Table[s[n], {n, 1, 200}]

from itertools import groupby
from sympy.ntheory import digits
def A043556(n): return len(list(groupby(digits(n,4)[1:]))) # Chai Wah Wu, Jul 13 2024

a(n) << log n. In particular, a(n) <= log(n)/log(4) + 1. - Charles R Greathouse IV, Jul 13 2024

Updated by Clark Kimberling, Feb 03 2018

A043567 Number of runs in base-15 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Clark Kimberling

Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences.

			For n = 226, its base-15 representation is "101" as 226 = 1*(15^2) + 0*(15^1) + 1*(15^0). "101" has three runs, thus a(226) = 3.
For n = 482, its base-15 representation is "222" as 482 = 2*(15^2) + 2*(15^1) + 2*(15^0). "222" has just one run, thus a(482) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A043289, A043542, A297783 (number of distinct runs), A297770.

Table[Length@ Split@ IntegerDigits[n, 15], {n, 0, 105}] (* Michael De Vlieger, Oct 10 2017 *)

(define (A043567 n) (let loop ((n n) (runs 1) (pd (modulo n 15))) (if (zero? n) runs (let ((d (modulo n 15))) (loop (/ (- n d) 15) (+ runs (if (not (= d pd)) 1 0)) d))))) ;; Antti Karttunen, Oct 10 2017

More terms from Antti Karttunen, Oct 10 2017

Updated by Clark Kimberling, Feb 04 2018

cp's OEIS Frontend

A351204 Number of integer partitions of n such that every permutation has all distinct runs.

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A350952 The smallest number whose binary expansion has exactly n distinct runs.

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A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

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A043562 Number of runs in base-10 representation of n.

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A043555 Number of runs in base-3 representation of n.

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A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

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A353930 Smallest number whose binary expansion has n distinct run-sums.

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