A374686 -id:A374686 - cp's OEIS frontend

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

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

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

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

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!GreaterEqual@@First/@Split[stc[#],Less]&]

cp's OEIS Frontend

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

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