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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..66.

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 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.

Mathematica

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

Crossrefs

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.

Sequence in context: A374038 A203076 A247180 * A317091 A048839 A360477

Adjacent sequences: A375395 A375396 A375397 * A375399 A375400 A375401

Keyword

nonn

Author

Gus Wiseman, Aug 16 2024

Status

approved