A044813 -id:A044813 - cp's OEIS frontend

A351201 Numbers whose multiset of prime factors has a permutation without all distinct runs.

12, 18, 20, 28, 36, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 120, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204, 207, 208, 212, 216

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

			The prime factors of 80 are {2,2,2,2,5} and the permutation (2,2,5,2,2) has runs (2,2), (5), and (2,2), which are not all distinct, so 80 is in the sequence. On the other hand, 24 has prime factors {2,2,2,3}, and all four permutations (3,2,2,2), (2,3,2,2), (2,2,3,2), (2,2,2,3) have distinct runs, so 24 is not in the sequence.
The terms and their prime indices begin:
     12: (2,1,1)         76: (8,1,1)        132: (5,2,1,1)
     18: (2,2,1)         80: (3,1,1,1,1)    140: (4,3,1,1)
     20: (3,1,1)         84: (4,2,1,1)      144: (2,2,1,1,1,1)
     28: (4,1,1)         90: (3,2,2,1)      147: (4,4,2)
     36: (2,2,1,1)       92: (9,1,1)        148: (12,1,1)
     44: (5,1,1)         98: (4,4,1)        150: (3,3,2,1)
     45: (3,2,2)         99: (5,2,2)        153: (7,2,2)
     48: (2,1,1,1,1)    100: (3,3,1,1)      156: (6,2,1,1)
     50: (3,3,1)        108: (2,2,2,1,1)    162: (2,2,2,2,1)
     52: (6,1,1)        112: (4,1,1,1,1)    164: (13,1,1)
     60: (3,2,1,1)      116: (10,1,1)       168: (4,2,1,1,1)
     63: (4,2,2)        117: (6,2,2)        171: (8,2,2)
     68: (7,1,1)        120: (3,2,1,1,1)    172: (14,1,1)
     72: (2,2,1,1,1)    124: (11,1,1)       175: (4,3,3)
     75: (3,3,2)        126: (4,2,2,1)      176: (5,1,1,1,1)

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

The version for run-lengths instead of runs is A024619.
These permutations are counted by A351202.
These rank the partitions counted by A351203, complement A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A283353 counts normal multisets with a permutation w/o all distinct runs.
A297770 counts distinct runs in binary expansion.
A333489 ranks anti-runs, complement A348612.
A351014 counts distinct runs in standard compositions, firsts A351015.
A351291 ranks compositions without all distinct runs.
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.
Cf. A001055, A001221, A001222, A061395, A098859, A106356, A106529.

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

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

Original entry on oeis.org

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

A044817 Positive integers having distinct base-6 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 36, 42, 43, 44, 45, 46, 47, 50, 57, 64, 71, 72, 79, 84, 85, 86, 87, 88, 89, 93, 100, 107, 108, 115, 122, 126, 127, 128, 129, 130, 131, 136, 143, 144, 151, 158, 165, 168, 169, 170, 171, 172, 173, 179, 180, 187, 194, 201, 208, 210, 211

Offset: 1

Clark Kimberling

			93 = 233_6 is in the sequence as it has distinct run lengths of same digits (1, 2). - _David A. Corneth_, Jan 04 2021

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

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

filter:= proc(n) local L,R,s,t,i;
  L:= convert(n,base,6);
  R:= NULL; t:= 1; s:= L[1];
  for i from 2 to nops(L) do
    if L[i] <> s then
      R:= R, t; t:= 1; s:= L[i]
    else
      t:= t+1
    fi
  od;
  R:= R, t;
  nops([R]) = nops({R})
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 17 2018

A044818 Positive integers having distinct base-7 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 49, 56, 57, 58, 59, 60, 61, 62, 65, 73, 81, 89, 97, 98, 106, 112, 113, 114, 115, 116, 117, 118, 122, 130, 138, 146, 147, 155, 163, 168, 169, 170, 171, 172, 173, 174, 179, 187, 195, 196, 204, 212, 220, 224, 225, 226, 227, 228

Offset: 1

Clark Kimberling

			40=55_7 has a single run length of 2 and is in the sequence. 211=421_7 has three runs of length 1 as is not in the sequence. - _R. J. Mathar_, Jan 18 2018

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

rlset := proc(L::list)
    local lset,rl,i ;
    lset := [] ;
    rl := 1 ;
    for i from 2 to nops(L) do
        if op(i,L) = op(i-1,L) then
            rl := rl+1 ;
        else
            lset := [op(lset),rl] ;
            rl := 1;
        end if;
    end do:
    lset := [op(lset),rl] ;
end proc:
isA044818 := proc(n)
    local dgs,rl;
    dgs := convert(n,base,7) ;
    rl := rlset(dgs) ;
    if nops(rl) = nops( convert(rl,set)) then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 400 do
    if isA044818(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018

A044819 Positive integers having distinct base-8 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 64, 72, 73, 74, 75, 76, 77, 78, 79, 82, 91, 100, 109, 118, 127, 128, 137, 144, 145, 146, 147, 148, 149, 150, 151, 155, 164, 173, 182, 191, 192, 201, 210, 216, 217, 218, 219, 220, 221, 222, 223, 228, 237, 246, 255, 256

Offset: 1

Clark Kimberling

			222 = 336_8 has a run length of two and a run length of 1, which are distinct lengths, so 222 is in the sequence. - _R. J. Mathar_, Jan 18 2018

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

rlset := proc(L::list)
    local lset,rl,i ;
    lset := [] ;
    rl := 1 ;
    for i from 2 to nops(L) do
        if op(i,L) = op(i-1,L) then
            rl := rl+1 ;
        else
            lset := [op(lset),rl] ;
            rl := 1;
        end if;
    end do:
    lset := [op(lset),rl] ;
end proc:
isA044819 := proc(n)
    local dgs,rl;
    dgs := convert(n,base,8) ;
    rl := rlset(dgs) ;
    if nops(rl) = nops( convert(rl,set)) then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 400 do
    if isA044819(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018

A044820 Positive integers having distinct base-9 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 81, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 111, 121, 131, 141, 151, 161, 162, 172, 180, 181, 182, 183, 184, 185, 186, 187, 188, 192, 202, 212, 222, 232, 242, 243, 253, 263, 270, 271, 272, 273, 274, 275, 276

Offset: 1

Clark Kimberling

			242 = 288_9 is in the sequence as it has distinct run lengths of distinct digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

rlset := proc(L::list)
    local lset,rl,i ;
    lset := [] ;
    rl := 1 ;
    for i from 2 to nops(L) do
        if op(i,L) = op(i-1,L) then
            rl := rl+1 ;
        else
            lset := [op(lset),rl] ;
            rl := 1;
        end if;
    end do:
    lset := [op(lset),rl] ;
end proc:
isA044820 := proc(n)
    local dgs,rl;
    dgs := convert(n,base,9) ;
    rl := rlset(dgs) ;
    if nops(rl) = nops( convert(rl,set)) then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 400 do
    if isA044820(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018

A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 4, 6, 10, 11, 17, 19, 30, 36, 51, 61, 84, 96, 133, 160, 209, 253, 325, 393, 488, 598, 744

Offset: 0

Gus Wiseman, May 07 2025

The Heinz numbers of these partitions are the odd terms of A382912.

Also the number of integer partitions of n with no ones whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(4) = 1 through a(12) = 10 partitions:
  (22)  .  (222)  (322)  (332)   (333)   (622)    (443)    (444)
                         (422)   (522)   (3322)   (722)    (822)
                         (2222)  (3222)  (4222)   (3332)   (3333)
                                         (22222)  (4322)   (4332)
                                                  (5222)   (4422)
                                                  (32222)  (5322)
                                                           (6222)
                                                           (33222)
                                                           (42222)
                                                           (222222)

The complement without ones is counted by A383533.
The number of these families is A383706.
Allowing ones gives A383710 (ranks A382912), complement A383708 (ranks A382913).
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&pof[#]=={}&]],{n,0,15}]

A044814 Positive integers having distinct base-3 run lengths.

Original entry on oeis.org

1, 2, 4, 8, 9, 12, 13, 14, 17, 18, 22, 24, 25, 26, 27, 39, 40, 41, 53, 54, 67, 78, 79, 80, 81, 108, 117, 120, 121, 122, 125, 134, 161, 162, 202, 216, 229, 234, 238, 240, 241, 242, 243, 247, 251, 256, 269, 324, 325, 326, 337, 350, 352, 353, 355, 359, 360, 363, 364

Offset: 1

Clark Kimberling

			18 = 200_3 is in the sequence as it has distinct runs of same base 3 digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

A044815 Positive integers having distinct base-4 run lengths.

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 16, 20, 21, 22, 23, 26, 31, 32, 37, 40, 41, 42, 43, 47, 48, 53, 58, 60, 61, 62, 63, 64, 84, 85, 86, 87, 106, 127, 128, 149, 168, 169, 170, 171, 191, 192, 213, 234, 252, 253, 254, 255, 256, 320, 336, 340, 341, 342, 343, 346, 351, 362, 383, 426, 511

Offset: 1

Clark Kimberling

			31 = 133_4 is in the sequence as it has distinct run lengths of same digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

A044816 Positive integers having distinct base-5 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 18, 24, 25, 30, 31, 32, 33, 34, 37, 43, 49, 50, 56, 60, 61, 62, 63, 64, 68, 74, 75, 81, 87, 90, 91, 92, 93, 94, 99, 100, 106, 112, 118, 120, 121, 122, 123, 124, 125, 155, 156, 157, 158, 159, 187, 218, 249, 250, 281, 310, 311, 312, 313, 314, 343

Offset: 1

Clark Kimberling

			32 = 112_5 is in the sequence as it has distinct run lengths of same digits (2, 1). - _David A. Corneth_, Jan 04 2021

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

cp's OEIS Frontend

A351201 Numbers whose multiset of prime factors has a permutation without all distinct runs.

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

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A044817 Positive integers having distinct base-6 run lengths.

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Maple

A044818 Positive integers having distinct base-7 run lengths.

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Maple

A044819 Positive integers having distinct base-8 run lengths.

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Maple

A044820 Positive integers having distinct base-9 run lengths.

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Maple

A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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Mathematica

A044814 Positive integers having distinct base-3 run lengths.

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A044815 Positive integers having distinct base-4 run lengths.

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