A375134 -id:A375134 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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

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