cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Robert G. Wilson v, Sep 30 2009

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

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

A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 216, 224, 232, 236, 242
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2025

Keywords

Comments

First differs from A177425, A182854, A367589 in having 216.
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.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)
so 360 is in the sequence.
The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}

Crossrefs

For exactly one permutation we have A000961, counted by A000005.
For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
For at least one choice we have A351294, conjugate A381432, counted by A239455.
These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
For equal run-lengths we have A383089, positions of terms > 1 in A382857.
Partitions of this type are counted by A383111.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]

Formula

The complement is A000961 \/ A351293, counted by A000005 + A351295.

A351205 Numbers whose binary expansion does not have all distinct runs.

Original entry on oeis.org

5, 9, 10, 17, 18, 20, 21, 22, 26, 27, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 51, 53, 54, 58, 65, 66, 68, 69, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 99, 100, 101, 102, 105, 106, 107, 108, 109, 110, 117, 118, 119, 122, 129
Offset: 1

Views

Author

Gus Wiseman, Feb 07 2022

Keywords

Examples

			The terms together with their binary expansions begin:
      5:     101     41:  101001     74: 1001010
      9:    1001     42:  101010     75: 1001011
     10:    1010     43:  101011     76: 1001100
     17:   10001     45:  101101     77: 1001101
     18:   10010     46:  101110     80: 1010000
     20:   10100     51:  110011     81: 1010001
     21:   10101     53:  110101     82: 1010010
     22:   10110     54:  110110     83: 1010011
     26:   11010     58:  111010     84: 1010100
     27:   11011     65: 1000001     85: 1010101
     33:  100001     66: 1000010     86: 1010110
     34:  100010     68: 1000100     87: 1010111
     36:  100100     69: 1000101     89: 1011001
     37:  100101     72: 1001000     90: 1011010
     40:  101000     73: 1001001     91: 1011011
For example, 77 has binary expansion 1001101, with runs 1, 00, 11, 0, 1, which are not all distinct, so 77 is in the sequence.

Links

Crossrefs

Runs in binary expansion are counted by A005811, distinct A297770.
The complement is A175413, for run-lengths A044813.
The version for standard compositions is A351291, complement A351290.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A325545 counts compositions with distinct differences.
A333489 ranks anti-runs, complement A348612, counted by A003242.
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.
- 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. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.

Programs

  • Maple
    q:= proc(n) uses ListTools; (l-> is(nops(l)<>add(
      nops(i), i={Split(`=`, l, 1)}) +add(
      nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
    end:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 14 2022
  • Mathematica
    Select[Range[0,100],!UnsameQ@@Split[IntegerDigits[#,2]]&]
  • Python
    from itertools import groupby, product
def ok(n):
    runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
    return len(runs) > len(set(runs))
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Feb 09 2022
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