A375396 -id:A375396 - cp's OEIS frontend

A375398 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are distinct.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

First differs from A375402 in lacking 20.
An anti-run is a sequence with no adjacent equal parts.
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.
Note the prime factors can alternatively be taken in weakly decreasing order.

			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 not in the sequence.
The prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is in the sequence.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

A version for compositions is A374638, counted by A374518.
These are positions of strict rows in A375128, sums A374706, ranks A375400.
Partitions (or reversed partitions) of this type are counted by A375134.
For identical instead of distinct we have A375396, counted by A115029.
The complement is A375399, counted by A375404.
For maxima instead of minima we have A375402, counted by A375133.
The complement for maxima 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. A034296, A036785, A046660, A065200, A065201, A358836, A374632, A374761, A374767, A374768, A375136.

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

A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 169, 171

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

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.
Note the prime factors can alternatively be taken in weakly decreasing order.

			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 prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is not in the sequence.
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    20: {1,1,3}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    44: {1,1,5}
    45: {2,2,3}
    48: {1,1,1,1,2}

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The complement for compositions is A374638, counted by A374518.
A version for compositions is A374639, counted by A374678.
Positions of non-strict rows in A375128, sums A374706, ranks A375400.
For identical instead of strict we have A375397, counted by A375405.
The complement is A375398, counted by A375134.
The complement for maxima instead of minima is A375402, counted by A375133.
For maxima instead of minima we have A375403, counted by A375401.
Partitions (or reversed partitions) of this type are counted by A375404.
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. A034296, A046660, A065200, A065201, A115029, A279790, A374632, A374761, A375136, A375396.

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

A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 33, 48, 63, 88, 116, 157, 204, 272, 349, 456, 581, 749, 946, 1205, 1511, 1904, 2371, 2960, 3661, 4538, 5577, 6862, 8389, 10257, 12472, 15164, 18348, 22192, 26731, 32177, 38593, 46254, 55256, 65952, 78500, 93340, 110706

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

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 partition y = (3,2,2,1) has maximal ant-runs ((3,2),(2,1)), with maxima (3,2), so y is not counted under a(8).
The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (2111)   (222)     (2221)     (332)
                       (11111)  (2211)    (4111)     (2222)
                                (3111)    (22111)    (3311)
                                (21111)   (31111)    (5111)
                                (111111)  (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

For identical instead of distinct we have A239955, ranks A073492.
The complement is counted by A375133, ranks A375402.
The complement for minima instead of maxima is A375134, ranks A375398.
These partitions have Heinz numbers A375403.
For minima instead of maxima we have A375404, ranks A375399.
The reverse for identical instead of distinct is A375405, ranks A375397.
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. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 6, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 12, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 10, 3, 4, 53, 18, 5, 8, 3, 2, 59, 4, 61, 2, 9, 64, 5, 2, 67, 4, 3, 2, 71, 24, 73, 2, 15, 4, 7

Offset: 1

Gus Wiseman, Aug 17 2024

nonn

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.
An anti-run is a sequence with no adjacent equal parts. 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.
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 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.

bigomega is A001222(a(n)) = A375136(n).
Least prime factor is A020639(a(n)) = A020639(n).
Least prime index is A055396(a(n)) = A055396(n).
Heinz weights are A056239(a(n)) = A374706(n).
The greatest prime index A061395(a(n)) is the maximum of row n of A375128.
Firsts for omega (except first term) are half A061742.
Prime indices A112798(a(n)) are row n of A375128.
Positions of prime-powers are A375396, counted by A115029.
Positions of squarefree numbers are A375398, counted by A375134.
A000041 counts integer partitions, strict A000009.
A027748 lists distinct prime factors, sum A008472.
A304038 lists distinct prime indices, sum A066328.
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. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.

Table[Times@@Prime/@If[n==1,{},Min /@ Split[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],UnsameQ]],{n,100}]

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

cp's OEIS Frontend

A375398 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are distinct.

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A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

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A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

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A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

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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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