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

Author

Gus Wiseman, Aug 16 2024

Keywords

• nonn

Comments

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

Examples

			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}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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.

Programs

Mathematica

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

PARI

is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) > e[1], 0); \\ Amiram Eldar, Oct 26 2024

Extensions

Name edited by Peter Munn, May 08 2025