A165413 -id:A165413 - cp's OEIS frontend

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

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

A005811 Number of runs in binary expansion of n (n>0); number of 1's in Gray code for n.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5

Offset: 0

N. J. A. Sloane, Jeffrey Shallit, Simon Plouffe

Starting with a(1) = 0 mirror all initial 2^k segments and increase by one.
a(n) gives the net rotation (measured in right angles) after taking n steps along a dragon curve. - Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
This sequence generates A082410: (0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...) and A014577; identical to the latter except starting 1, 1, 0, ...; by writing a "1" if a(n+1) > a(n); if not, write "0". E.g., A014577(2) = 0, since a(3) < a(2), or 1 < 2. - Gary W. Adamson, Sep 20 2003
Starting with 1 = partial sums of A034947: (1, 1, -1, 1, 1, -1, -1, 1, 1, 1, ...). - Gary W. Adamson, Jul 23 2008
The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.
Can be used as a binomial transform operator: Let a(n) = the n-th term in any S(n); then extract 2^k strings, adding the terms. This results in the binomial transform of S(n). Say S(n) = 1, 3, 5, ...; then we obtain the strings: (1), (3, 1), (3, 5, 3, 1), (3, 5, 7, 5, 3, 5, 3, 1), ...; = the binomial transform of (1, 3, 5, ...) = (1, 4, 12, 32, 80, ...). Example: the 8-bit string has a sum of 32 with a distribution of (1, 3, 3, 1) or one 1, three 3's, three 5's, and one 7; as expected. - Gary W. Adamson, Jun 21 2012
Considers all positive odd numbers as nodes of a graph. Two nodes are connected if and only if the sum of the two corresponding odd numbers is a power of 2. Then a(n) is the distance between 2n + 1 and 1. - Jianing Song, Apr 20 2019

			Considered as a triangle with 2^k terms per row, the first few rows are:
  1
  2, 1
  2, 3, 2, 1
  2, 3, 4, 3, 2, 3, 2, 1
  ...
The n-th row becomes right half of next row; left half is mirrored terms of n-th row increased by one. - _Gary W. Adamson_, Jun 20 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook).
J.-P. Allouche, G.-N. Han and J. Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Danielle Cox and Karyn McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105-113.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted with addendum in Donald E. Knuth, Selected Papers on Fun and Games, 2010, pages 571-614. Equation 3.2 g(n) = a(n-1).
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
P. Flajolet et al., Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.
P. Flajolet and Lyle Ramshaw, A note on Gray code and odd-even merge, SIAM J. Comput. 9 (1980), 142-158.
Sara Kropf and Stephan Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016.
Sara Kropf and S. Wagner, On q-Quasiadditive and q-Quasimultiplicative Functions, arXiv preprint arXiv:1608.03700 [math.CO], 2016.
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Helmut Prodinger and Friedrich J. Urbanek, Infinite 0-1-Sequences Without Long Adjacent Identical Blocks, Discrete Mathematics, volume 28, issue 3, 1979, pages 277-289. Also first author's copy. Their "variation" v(k) at definition 3.4 is a(k).
Jeffrey Shallit, The mathematics of Per Noergaard's rhythmic infinity system, Fib. Q., 43 (2005), 262-268.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Index entries for "core" sequences.
Index entries for sequences related to binary expansion of n.

Cf. A037834 (-1), A088748 (+1), A246960 (mod 4), A034947 (first differences), A000975 (indices of record highs), A173318 (partial sums).
Cf. A056539, A014707, A014577, A082410, A030308, A090079, A044813, A165413, A226227, A226228, A226229.
Partial sums of A112347. Recursion depth of A035327.
Cf. A000120, A003188.

import Data.List (group)
a005811 0 = 0
a005811 n = length $ group $ a030308_row n
a005811_list = 0 : f [1] where
   f (x:xs) = x : f (xs ++ [x + x `mod` 2, x + 1 - x `mod` 2])
-- Reinhard Zumkeller, Feb 16 2013, Mar 07 2011

A005811 := proc(n)
    local i, b, ans;
    if n = 0 then
        return 0 ;
    end if;
    ans := 1;
    b := convert(n, base, 2);
    for i from nops(b)-1 to 1 by -1 do
        if b[ i+1 ]<>b[ i ] then
            ans := ans+1
        fi
    od;
    return ans ;
end proc:
seq(A005811(i), i=1..50) ;
# second Maple program:
a:= n-> add(i, i=Bits[Split](Bits[Xor](n, iquo(n, 2)))):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 01 2023

Table[ Length[ Length/@Split[ IntegerDigits[ n, 2 ] ] ], {n, 1, 255} ]
a[n_] := DigitCount[BitXor[n, Floor[n/2]], 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 11 2024 *)

a(n)=sum(k=1,n,(-1)^((k/2^valuation(k,2)-1)/2))

a(n)=if(n<1,0,a(n\2)+(a(n\2)+n)%2) \\ Benoit Cloitre, Jan 20 2014

a(n) = hammingweight(bitxor(n, n>>1));  \\ Gheorghe Coserea, Sep 03 2015

def a(n): return bin(n^(n>>1))[2:].count("1") # Indranil Ghosh, Apr 29 2017

a(2^k + i) = a(2^k - i + 1) + 1 for k >= 0 and 0 < i <= 2^k. - Reinhard Zumkeller, Aug 14 2001
a(2n+1) = 2a(n) - a(2n) + 1, a(4n) = a(2n), a(4n+2) = 1 + a(2n+1).
a(j+1) = a(j) + (-1)^A014707(j). - Christopher Hendrie (hendrie(AT)acm.org), Sep 11 2002
G.f.: (1/(1-x)) * Sum_{k>=0} x^2^k/(1+x^2^(k+1)). - Ralf Stephan, May 02 2003
Delete the 0, make subsets of 2^n terms; and reverse the terms in each subset to generate A088696. - Gary W. Adamson, Oct 19 2003
a(0) = 0, a(2n) = a(n) + [n odd], a(2n+1) = a(n) + [n even]. - Ralf Stephan, Oct 20 2003
a(n) = Sum_{k=1..n} (-1)^((k/2^A007814(k)-1)/2) = Sum_{k=1..n} (-1)^A025480(k-1). - Ralf Stephan, Oct 29 2003
a(n) = A069010(n) + A033264(n). - Ralf Stephan, Oct 29 2003
a(0) = 0 then a(n) = a(floor(n/2)) + (a(floor(n/2)) + n) mod 2. - Benoit Cloitre, Jan 20 2014
a(n) = A037834(n) + 1.
a(n) = A000120(A003188(n)). - Amiram Eldar, Jul 11 2024

Additional description from Wouter Meeussen

A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Gus Wiseman, Feb 15 2022

nonn

First differs from A130091 (Wilf partitions) in having 216.
See A239455 for the definition of Look-and-Say partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()            20: (3,1,1)         47: (15)
      2: (1)           23: (9)             48: (2,1,1,1,1)
      3: (2)           24: (2,1,1,1)       49: (4,4)
      4: (1,1)         25: (3,3)           50: (3,3,1)
      5: (3)           27: (2,2,2)         52: (6,1,1)
      7: (4)           28: (4,1,1)         53: (16)
      8: (1,1,1)       29: (10)            54: (2,2,2,1)
      9: (2,2)         31: (11)            56: (4,1,1,1)
     11: (5)           32: (1,1,1,1,1)     59: (17)
     12: (2,1,1)       37: (12)            61: (18)
     13: (6)           40: (3,1,1,1)       63: (4,2,2)
     16: (1,1,1,1)     41: (13)            64: (1,1,1,1,1,1)
     17: (7)           43: (14)            67: (19)
     18: (2,2,1)       44: (5,1,1)         68: (7,1,1)
     19: (8)           45: (3,2,2)         71: (20)
For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).

The Wilf case (distinct multiplicities) is A130091, counted by A098859.
The complement of the Wilf case is A130092, counted by A336866.
These partitions appear to be counted by A239455.
A variant for runs is A351201, counted by A351203 (complement A351204).
The complement is A351295, counted by A351293.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of run-lengths in binary expansion, for all runs A297770.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]!={}&]

Name edited by Gus Wiseman, Aug 13 2025

A044813 Positive integers having distinct base-2 run lengths.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 14, 15, 16, 24, 28, 30, 31, 32, 35, 39, 48, 49, 55, 57, 59, 60, 62, 63, 64, 67, 79, 96, 97, 111, 112, 120, 121, 123, 124, 126, 127, 128, 131, 135, 143, 159, 192, 193, 223, 224, 225, 239, 241, 247, 248, 249, 251, 252, 254, 255, 256, 259, 263

Offset: 1

Clark Kimberling

A005811(a(n)) = A165413(a(n)). - Reinhard Zumkeller, Mar 02 2013
From Emeric Deutsch, Jan 25 2018: (Start)
Also, the indices of the compositions that have distinct parts. For the definition of the index of a composition see A298644. For example, 223 is in the sequence since its binary form is 11011111 and the composition [2,1,5] has distinct parts. 100 is not in the sequence since its binary form is 1100100 and the parts of the composition [2,2,1,2] are not distinct.
The command c(n) from the Maple program yields the composition having index n. (End)

Reinhard Zumkeller and Gheorghe Coserea, Table of n, a(n) for n = 1..20000 (first 5000 terms from Reinhard Zumkeller)

Cf. A030308, A298644.

import Data.List (group, nub)
a044813 n = a044813_list !! (n-1)
a044813_list = filter p [1..] where
   p x = nub xs == xs where
         xs = map length $ group $ a030308_row x
-- Reinhard Zumkeller, Mar 02 2013

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 300 do if nops(convert(c(n), set)) = nops(c(n)) then A := `union`(A, {n}) else end if end do: A; # most of the Maple program is due to W. Edwin Clark. - Emeric Deutsch, Jan 25 2018

f[n_] := Unequal@@Length/@Split[IntegerDigits[n,2]]; Select[Range[300],f] (* Ray Chandler, Oct 21 2011 *)

is(n) = {
  my(v = 0, hist = vector(1 + logint(n+1, 2)));
  while(n != 0,
        v = valuation(n, 2); n >>= v; n++;
        hist[v+1]++; if (hist[v+1] >= 2, return(0));
        v = valuation(n, 2); n >>= v; n--;
        hist[v+1]++; if (hist[v+1] >= 2, return(0)));
  return(1);
};
seq(n) = {
  my(k = 1, top = 0, v = vector(n));
  while(top < n, if (is(k), v[top++] = k); k++);
  return(v);
};
seq(59) \\ Gheorghe Coserea, Nov 02 2015

from itertools import groupby
def ok(n):
  runlengths = [len(list(g)) for k, g in groupby(bin(n)[2:])]
  return len(runlengths) == len(set(runlengths))
print([i for i in range(1, 264) if ok(i)]) # Michael S. Branicky, Jan 04 2021

a(Sum_{k=0..n} A032020(k)) = 2^n, for n>1. - Gheorghe Coserea, May 30 2017

Extended by Ray Chandler, Oct 21 2011

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

A353833 Numbers whose multiset of prime indices has all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

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 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.

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

For parts instead of run-sums we have A000961, counted by A000005.
For run-lengths instead of run-sums we have A072774, counted by A047966.
These partitions are counted by A304442.
These are the positions of powers of primes in A353832.
The restriction to nonprimes is A353834.
For distinct instead of equal run-sums we have A353838, counted by A353837.
The version for compositions is A353848, counted by A353851.
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 deal with iterated run-sums for partitions.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A007947, A071625, A073093, A181819, A238279, A304660, A323014, A333755, A353839.

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

A353838 Numbers whose prime indices have all distinct run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71

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 prime indices of 180 are {1,1,2,2,3}, with run-sums (2,4,3), so 180 is in the sequence.
The prime indices of 315 are {2,2,3,4}, with run-sums (4,3,4), so 315 is not in the sequence.

The version for all equal run-sums is A353833, counted by A304442.
These partitions are counted by A353837.
The complement is A353839.
The version for compositions is A353852, counted by A353850.
The greatest run-sum is given by A353862, least A353931.
The weak case is A353866, counted by A353864.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A002110, A071625, A073093, A118914, A124010, A353834, A353861, A353867.

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

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 30 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 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.

			Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.

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

Counting repeated runs also gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of run-sums we have A351014 (firsts A351015).
A version for partitions is A353835, weak A353861.
Positions of 1's are A353848, counted by A353851.
The version for binary expansion is A353929 (firsts A353930).
The run-sums themselves are listed by A353932, with A353849 distinct terms.
For distinct run-lengths instead of run-sums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct run-lengths in binary expansion.
A297770 counts distinct runs in binary expansion, firsts A350952.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Total/@Split[stc[n]]]],{n,0,100}]

A353839 Numbers whose prime indices do not have all distinct run-sums.

Original entry on oeis.org

12, 40, 60, 63, 84, 112, 120, 126, 132, 144, 156, 204, 228, 252, 276, 280, 300, 315, 325, 336, 348, 351, 352, 360, 372, 420, 440, 444, 492, 504, 516, 520, 560, 564, 588, 630, 636, 650, 660, 675, 680, 693, 702, 708, 720, 732, 760, 780, 804, 819, 832, 840, 852

Offset: 1

Gus Wiseman, Jun 04 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.

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 together with their prime indices begin:
   12: {1,1,2}
   40: {1,1,1,3}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  280: {1,1,1,3,4}
  300: {1,1,2,3,3}
  315: {2,2,3,4}

For equal run-sums we have A353833, counted by A304442, nonprime A353834.
The complement is A353838, counted by A353837.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353862 gives the greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002110, A071625, A073093, A116608, A118914, A124010, A353861, A353867.

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

A353861 Number of distinct weak run-sums of the prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 5, 2, 4, 2, 4, 3, 3, 2, 4, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 4, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 5, 3, 4, 3, 4, 2, 5, 3, 5, 3, 3, 2, 4, 2, 3, 3, 7, 3, 4, 2, 4, 3, 4, 2, 5, 2, 3, 4, 4, 3, 4, 2, 5, 5, 3, 2, 4, 3, 3, 3, 5, 2, 5, 3, 4, 3, 3, 3, 6, 2, 4, 4, 5, 2, 4, 2, 5, 4, 3, 2, 5

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.

A weak run-sum of a sequence is the sum of any consecutive constant subsequence.

			The prime indices of 72 are {1,1,1,2,2}, with weak runs {}, {1}, {1,1}, {1,1,1}, {2}, {2,2}, which have sums 0, 1, 2, 3, 2, 4, of which 5 are distinct, so a(72) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of 2's are A000040.
Positions of first appearances are A000079.
The strong version is A353835, firsts A002110.
Partitions with distinct run-sums are ranked by A353838, counted by A353837.
The strong version for compositions is A353849.
The greatest run-sum is given by A353862, least A353931.
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.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353834, A353839, A353866, A353867, A353930.

Table[Length[Union@@Cases[FactorInteger[n],{p_,k_}:>Range[0,k]*PrimePi[p]]],{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); };
A353861(n) = if(1==n,n,my(pruns = pis_to_runs(n), runsum = 0, runsums = List([])); for(i=1,#pruns, listput(runsums, runsum); if((i>1) && pruns[i] == pruns[i-1], runsum += pruns[i], runsum = pruns[i])); listput(runsums, runsum); #Set(runsums)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

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

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A005811 Number of runs in binary expansion of n (n>0); number of 1's in Gray code for n.

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A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

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A044813 Positive integers having distinct base-2 run lengths.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A353833 Numbers whose multiset of prime indices has all equal run-sums.

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A353838 Numbers whose prime indices have all distinct run-sums.

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