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.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..59.

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

Example

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}

Mathematica

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

Crossrefs

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.

Sequence in context: A057109 A369639 A069189 * A371601 A283050 A069168

Adjacent sequences: A375396 A375397 A375398 * A375400 A375401 A375402

Keyword

nonn

Author

Gus Wiseman, Aug 16 2024

Status

approved