A375136 -id:A375136 - cp's OEIS frontend

A384881 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive parts decreasing by 1.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 2, 2, 2, 0, 1, 0, 2, 3, 3, 2, 0, 1, 0, 2, 5, 3, 2, 2, 0, 1, 0, 1, 8, 4, 4, 2, 2, 0, 1, 0, 3, 5, 10, 4, 3, 2, 2, 0, 1, 0, 2, 9, 9, 9, 5, 3, 2, 2, 0, 1, 0, 2, 11, 13, 9, 9, 4, 3, 2, 2, 0, 1

Offset: 0

Gus Wiseman, Jun 25 2025

			The partition (5,4,2,1,1) has maximal runs ((5,4),(2,1),(1)) so is counted under T(13,3) = 23.
Row n = 9 counts the following partitions:
  9    63    333    6111    33111   411111   3111111   111111111
  54   72    441    22221   51111   2211111  21111111
  432  81    522    42111   222111
       621   531    321111
       3321  711
             3222
             4221
             4311
             5211
             32211
Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  1  3  0  1
  0  2  2  2  0  1
  0  2  3  3  2  0  1
  0  2  5  3  2  2  0  1
  0  1  8  4  4  2  2  0  1
  0  3  5 10  4  3  2  2  0  1
  0  2  9  9  9  5  3  2  2  0  1
  0  2 11 13  9  9  4  3  2  2  0  1
  0  2 13 15 17  8 10  4  3  2  2  0  1
  0  2 14 23 16 17  8  9  4  3  2  2  0  1
  0  2 16 26 26 19 16  9  9  4  3  2  2  0  1
  0  4 13 37 32 26 19 16  8  9  4  3  2  2  0  1

John Tyler Rascoe, Rows n = 0..100, flattened

Row sums are A000041.
Column k = 1 is A001227.
For distinct parts instead of maximal runs we have A116608.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
Partitions with distinct runs of this type are counted by A384882, gapless A384884.
For prime indices see A385213, A287170, A066205, A356229.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A000217, A008284, A047966, A047993, A098859, A325325, A375136, A384887.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1==#2+1&]]==k&]],{n,0,10},{k,0,n}]

tri(n) = {(n*(n+1)/2)}
B_list(N) = {my(v = vector(N, i, 0)); v[1] = q*t; for(m=2,N, v[m] = t * (q^tri(m) + sum(i=1,m-1, q^tri(i) * v[m-i] * (q^((m-i)*(i-1))/(1 - q^(m-i)) - q^((m-i)*i) + O('q^(N-tri(i)+1)))))); v}
A_qt(max_row) = {my(N = max_row+1, B = B_list(N), g = 1 + sum(m=1,N, B[m]/(1 - q^m)) + O('q^(N+1))); vector(N, n, Vecrev(polcoeff(g, n-1)))} \\ John Tyler Rascoe, Aug 18 2025

G.f.: 1 + Sum_{m>0} B(m,q,t)/(1 - q^m) where B(m,q,t) = t * (q^tri(m) + Sum_{i=1..m-1} q^tri(i) * B(m-i,q,t) * ((q^((m-i)*(i-1))/(1 - q^(m-i))) - q^((m-i)*i))) and tri(n) = A000217(n). - John Tyler Rascoe, Aug 18 2025

A375402 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) have distinct maxima.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89

Offset: 1

Gus Wiseman, Aug 14 2024

nonn

First differs from A349810 in lacking 150.
An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
The partitions with these Heinz numbers are those with (1) no part appearing more than twice and (2) the greatest part appearing only once.
Note the prime factors can alternatively be written in weakly decreasing order.
How is does the sequence relate to A317092? - R. J. Mathar, Aug 20 2024

			The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is not in the sequence.
The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is in the sequence.
The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is not in the sequence.

For identical instead of distinct we have A065200, complement A065201.
A version for compositions (instead of partitions) is A374767.
Partitions of this type are counted by A375133.
For minima instead of maxima we have A375398, counted by A375134.
The complement for minima is A375399, counted by A375404.
The complement is A375403, counted by A375401.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A046660, A066328, A358830, A374632, A374706, A374768, A375128, A375136, A375396, A375400.

Select[Range[150],UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375397 Numbers divisible by the square of some prime factor other than the least. Non-hooklike numbers.

Original entry on oeis.org

18, 36, 50, 54, 72, 75, 90, 98, 100, 108, 126, 144, 147, 150, 162, 180, 196, 198, 200, 216, 225, 234, 242, 245, 250, 252, 270, 288, 294, 300, 306, 324, 338, 342, 350, 360, 363, 375, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 486, 490, 500, 504, 507, 522

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

Contains no squarefree numbers A005117 or prime powers A000961, but some perfect powers A131605.
Also numbers k such that the minima of the maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not identical. Here, an anti-run is a sequence with no adjacent equal parts, and the minima of the maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each. Note the prime factors can alternatively be taken in weakly decreasing order.
Includes all terms of A036785 = non-products of a squarefree number and a prime power.
The asymptotic density of this sequence is 1 - (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q)) = 0.11514433883... . - Amiram Eldar, Oct 26 2024

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is in the sequence.
The terms together with their prime indices begin:
    18: {1,2,2}
    36: {1,1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    75: {2,3,3}
    90: {1,2,2,3}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   126: {1,2,2,4}
   144: {1,1,1,1,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..10000

A superset of A036785.
The complement for maxima is A065200, counted by A034296.
For maxima instead of minima we have A065201, counted by A239955.
A version for compositions is A374520, counted by A374640.
Also positions of non-constant rows in A375128, sums A374706, ranks A375400.
The complement is A375396, counted by A115029.
The complement for distinct minima is A375398, counted by A375134.
For distinct instead of identical minima we have A375399, counts A375404.
Partitions of this type are counted by A375405.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.

Select[Range[100],!SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) > e[1], 0); \\ Amiram Eldar, Oct 26 2024

Name edited by Peter Munn, May 08 2025

A375403 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.

Original entry on oeis.org

4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 48, 49, 50, 54, 56, 64, 72, 75, 80, 81, 88, 96, 98, 100, 104, 108, 112, 120, 121, 125, 128, 135, 136, 144, 147, 150, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 242, 243, 245

Offset: 1

Gus Wiseman, Aug 15 2024

nonn

First differs from A299117 in having 150.
An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
The partitions with these Heinz numbers are those with (1) some part appearing more than twice or (2) the greatest part appearing more than once.
Note the prime factors can alternatively be written in weakly decreasing order.

			The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is in the sequence.
The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is not in the sequence.
The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is in the sequence.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}

For identical instead of distinct we have A065201, complement A065200.
The complement for minima is A375398, counted by A375134.
For minima instead of maxima we have A375399, counted by A375404.
Partitions of this type are counted by A375401.
The complement is A375402, counted by A375133.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A046660, A066328, A358836, A374632, A374706, A374768, A374767, A375128, A375136, A375396, A375400.

Select[Range[150],!UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375405 Number of integer partitions of n with a repeated part other than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 13, 20, 29, 42, 62, 83, 117, 158, 214, 283, 377, 488, 641, 823, 1058, 1345, 1714, 2154, 2713, 3387, 4222, 5230, 6474, 7959, 9782, 11956, 14591, 17737, 21529, 26026, 31422, 37811, 45425, 54418, 65097, 77652, 92510, 109943, 130468

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

Also partitions whose minima of maximal anti-runs are not identical. An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

			The a(0) = 0 through a(10) = 13 partitions:
  .  .  .  .  .  (221)  (2211)  (331)    (332)     (441)      (442)
                                (2221)   (3221)    (3321)     (3322)
                                (22111)  (3311)    (4221)     (3331)
                                         (22211)   (22221)    (4411)
                                         (221111)  (32211)    (5221)
                                                   (33111)    (32221)
                                                   (222111)   (33211)
                                                   (2211111)  (42211)
                                                              (222211)
                                                              (322111)
                                                              (331111)
                                                              (2221111)
                                                              (22111111)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from John Tyler Rascoe)
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The complement for maxima instead of minima is A034296.
The complement is counted by A115029, ranks A375396.
For maxima instead of minima we have A239955, ranks A073492.
These partitions have ranks A375397.
For distinct instead of identical we have A375404, ranks A375399.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A046660, A065200, A065201, A141199, A358836, A374704, A375133, A375134, A375136, A375401.

Table[Length[Select[IntegerPartitions[n], !SameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]
- or -
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@DeleteCases[#,Min@@#]&]],{n,0,30}]

A_x(N) = {my(x='x+O('x^N), f=sum(i=1,N,sum(j=i+1,N-i, ((x^(i+(2*j)))/(1-x^i))*prod(k=i+1,N-i-(2*j), if(kJohn Tyler Rascoe, Aug 21 2024

G.f.: Sum_{i>0} (Sum_{j>i} ( (x^(i+(2*j)))/(1-x^i) * Product_{k>=i} (1-[kJohn Tyler Rascoe, Aug 21 2024

A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163

Offset: 1

Gus Wiseman, Jul 04 2025

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.

These are also numbers with the same number of adjacent equal prime indices as adjacent unequal prime indices.

			The prime indices of 2640 are {1,1,1,1,2,3,5}, with 4 distinct parts {1,2,3,5} and 4 maximal anti-runs ((1),(1),(1),(2,3,5)), so 2640 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  28: {1,1,4}
  29: {10}
  31: {11}
  37: {12}
  41: {13}
  43: {14}
  44: {1,1,5}
  45: {2,2,3}
  47: {15}

The LHS is the rank statistic A001221, triangle counted by A116608.
The RHS is the rank statistic A375136, triangle counted by A133121.
These partitions are counted by A385574.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length, ranks A106529.
A356235 counts partitions with a neighborless singleton, ranks A356237.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by maximal anti-runs, for partitions A268193, strict A384905.
A385572 counts subsets with the same number of runs as anti-runs, ranks A385575.
Cf. A044813, A046660, A210034, A297155, A356226, A361205, A384889, A385213.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||PrimeNu[#]==Length[Split[prix[#],UnsameQ]]&]

A001221(a(n)) = A375136(a(n)).

cp's OEIS Frontend

A384881 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive parts decreasing by 1.

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A375402 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) have distinct maxima.

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A375397 Numbers divisible by the square of some prime factor other than the least. Non-hooklike numbers.

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A375403 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.

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A375405 Number of integer partitions of n with a repeated part other than the least.

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A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

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Mathematica

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