A165413 -id:A165413 - cp's OEIS frontend

A353835 Number of distinct run-sums of the prime indices of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

Offset: 1

Gus Wiseman, May 23 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 prime indices of 3780 are {1,1,2,2,2,3,4}, with distinct run-sums {2,3,4,6}, so a(3780) = 4.
The prime indices of 8820 are {1,1,2,2,3,4,4}, with distinct run-sums {2,3,4,8}, so a(8820) = 4.
The prime indices of 13860 are {1,1,2,2,3,4,5}, with distinct run-sums {2,3,4,5}, so a(13860) = 4.
The prime indices of 92400 are {1,1,1,1,2,3,3,4,5}, with distinct run-sums {2,4,5,6}, so a(92400) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A002110.
A version for binary expansion is A165413.
Positions of 0's and 1's are A353833, nonprime A353834, counted by A304442.
The case of all distinct run-sums is ranked by A353838, counted by A353837.
The version for compositions is A353849.
The weak version is A353861.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353839, A353848, A353850, A353852, A353867.

Table[Length[Union[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
A353835(n) = omega(A353832(n)); \\ Antti Karttunen, Jan 20 2025

a(n) = A001221(A353832(n)). [From formula section of A353832] - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A351292 Number of patterns of length n with all distinct run-lengths.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 57, 61, 109, 161, 1265, 1317, 2469, 3577, 5785, 43901, 47165, 86337, 127665, 204853, 284197, 2280089, 2398505, 4469373, 6543453, 10570993, 14601745, 22502549, 159506453, 171281529, 314077353, 462623821, 742191037, 1031307185, 1580543969, 2141246229

Offset: 0

Gus Wiseman, Feb 10 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(5) = 9 patterns:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
              (1,1,2)  (1,1,1,2)  (1,1,1,1,2)
              (1,2,2)  (1,2,2,2)  (1,1,1,2,2)
              (2,1,1)  (2,1,1,1)  (1,1,2,2,2)
              (2,2,1)  (2,2,2,1)  (1,2,2,2,2)
                                  (2,1,1,1,1)
                                  (2,2,1,1,1)
                                  (2,2,2,1,1)
                                  (2,2,2,2,1)
The a(6) = 57 patterns grouped by sum:
  111111  111112  111122  112221  111223  111233  112333  122333
          111211  111221  122211  111322  111332  113332  133322
          112111  122111  211122  112222  112223  122233  221333
          211111  221111  221112  211222  113222  133222  223331
                                  221113  122222  211333  333122
                                  222112  211133  222133  333221
                                  222211  221222  222331
                                  223111  222113  233311
                                  311122  222122  331222
                                  322111  222221  332221
                                          222311  333112
                                          233111  333211
                                          311222
                                          322211
                                          331112
                                          332111

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for runs instead of run-lengths is A351200.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A165413 counts distinct run-lengths in binary expansion, runs A297770.
A345194 counts alternating patterns, up/down A350354.
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.
- A351202 = permutations of prime factors.
- A351638 = word structures.
Row sums of A350824.
Cf. A003242, A098504, A098859, A106356, A239455, A242882, A325545, A328592, A329740, A351014, A351293.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u,k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
seq(n)={my(u=P(n), c=poldegree(u[#u])); concat([1], sum(k=1, c, R(u, k)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 11 2022

From Andrew Howroyd, Feb 12 2022: (Start)
a(n) = Sum_{k=1..n} R(n,k)*(Sum_{r=k..n} binomial(r, k)*(-1)^(r-k)), where R(n,k) = Sum_{j=1..floor((sqrt(8*n+1)-1)/2)} k*(k-1)^(j-1) * j! * A008289(n,j).
G.f.: 1 + Sum_{r>=1} Sum_{k=1..r} R(k,x) * binomial(r, k)*(-1)^(r-k), where R(k,x) = Sum_{j>=1} k*(k-1)^(j-1) * j! * [y^j](Product_{k>=1} 1 + y*x^k).
(End)

Terms a(10) and beyond from Andrew Howroyd, Feb 11 2022

A353834 Nonprime numbers whose prime indices have all equal run-sums.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401

Offset: 1

Gus Wiseman, May 26 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    40: {1,1,1,3}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   112: {1,1,1,1,4}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).

For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
For compositions we have A353848\A000079, counted by A353851(n)-1.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.

Select[Range[100],!PrimeQ[#]&&SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue,1)))
A353848_list = list(islice(A353848_gen(),30)) # Chai Wah Wu, May 27 2022

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 2, 5, 0, 0, 0, 0, 5, 5, 1, 0, 0, 0, 0, 2, 12, 1, 0, 0, 0, 0, 0, 7, 12, 3, 0, 0, 0, 0, 0, 0, 3, 19, 8, 0, 0, 0, 0, 0, 0, 0, 5, 27, 9, 1, 0, 0, 0, 0, 0, 0, 0, 2, 33, 20, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, May 26 2022

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  4  1  0  0
  0  2  5  0  0  0
  0  5  5  1  0  0  0
  0  2 12  1  0  0  0  0
  0  7 12  3  0  0  0  0  0
  0  3 19  8  0  0  0  0  0  0
  0  5 27  9  1  0  0  0  0  0  0
  0  2 33 20  1  0  0  0  0  0  0  0
  0 13 28 34  2  0  0  0  0  0  0  0  0
  0  2 48 46  5  0  0  0  0  0  0  0  0  0
  0  5 65 51 14  0  0  0  0  0  0  0  0  0  0
  0  4 57 99 15  1  0  0  0  0  0  0  0  0  0  0
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (41111)     (611)
  (221111)    (3311)
  (11111111)  (4211)
              (5111)
              (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Counting distinct parts instead of run-sums gives A116608.
Column k = 1 is A304442, ranked by A353833 (nonprime A353834).
The rank statistic is A353835, weak A353861, for compositions A353849.
A275870 counts collapsible partitions, ranked by A300273.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
Cf. A008284, A047966, A071625, A165413, A175413, A325280, A333755.

Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Split[#]]]==k&]],{n,0,15},{k,0,n}]

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

A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93

Offset: 0

Gus Wiseman, May 15 2022

			The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

The non-recursive version is A353390, ranked by A353402.
The non-recursive consecutive version is A353392, ranked by A353432.
The non-recursive reverse version is A353403.
The unordered version is A353426, ranked by A353393 (nonprime A353389).
The consecutive version is A353430.
These compositions are ranked by A353431.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A329738 counts uniform compositions, partitions A047966.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yosQ]],{n,0,15}]

A353400 Number of integer compositions of n with all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 2, 4, 4, 5, 11, 11, 14, 27, 29, 37, 61, 72, 97, 147, 181, 246, 368, 470, 632, 914, 1198, 1611, 2286, 3018, 4079, 5709, 7619, 10329, 14333, 19258, 26142, 36069, 48688, 66114, 90800, 122913, 167020, 228735, 310167, 421708, 576499, 782803

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(7) = 1 through a(12) = 11 compositions:
  1111111   2222       333         22222        1112222       444
            11111111   111222      1111222      2222111       3333
                       222111      2221111      11111222      111333
                       111111111   1111111111   22211111      222222
                                                11111111111   333111
                                                              11112222
                                                              22221111
                                                              111111222
                                                              111222111
                                                              222111111
                                                              111111111111

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The = 2 version is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A078012, both A353428.
The version for partitions is A100405.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A274174 counts compositions with equal parts contiguous.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(21)-a(49) from Alois P. Heinz, May 17 2022

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 53.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

A353401 Number of integer compositions of n with all prime run-lengths.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 3, 6, 9, 10, 18, 27, 35, 54, 83, 107, 176, 242, 354, 515, 774, 1070, 1648, 2332, 3429, 4984, 7326, 10521, 15591, 22517, 32908, 48048, 70044, 101903, 149081, 216973, 316289, 461959, 672664, 981356, 1431256, 2086901, 3041577, 4439226, 6467735

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
  0   .  11   111   22   11111   33     11122     44       333
                                 222    22111     1133     11133
                                 1122   1111111   3311     33111
                                 2211             11222    111222
                                                  22211    222111
                                                  112211   1111122
                                                           1112211
                                                           1122111
                                                           2211111

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

The case of runs equal to 2 is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A023360, both A353429.
The version for partitions is A055923.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
The > 2 version is A353400, partitions A100405.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.
A005811 counts runs in binary expansion.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A167606 counts compositions with adjacent parts coprime.
A329738 counts uniform compositions, partitions A047966.
Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
     `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 18 2022

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

a(21)-a(45) from Alois P. Heinz, May 18 2022

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A353834 Nonprime numbers whose prime indices have all equal run-sums.

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A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

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

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A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

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