A375396 Numbers not divisible by the square of any prime factor except (possibly) the least. Hooklike numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Author

Gus Wiseman, Aug 16 2024

Keywords

• nonn

Comments

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 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.

The complement is a superset of A036785 = products of a squarefree number and a prime power.

The asymptotic density of this sequence is (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q)) = 0.884855661165... . - 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 not in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

The complement is a superset of A036785.

For maxima instead of minima we have A065200, counted by A034296.

The complement for maxima is A065201, counted by A239955.

Partitions of this type are counted by A115029.

A version for compositions is A374519, counted by A374517.

Also positions of identical rows in A375128, sums A374706, ranks A375400.

The complement is A375397, counted by A375405.

For distinct instead of identical minima we have A375398, counts A375134.

The complement for distinct minima is A375399, counted by A375404.

A000041 counts integer partitions, strict A000009.

A003242 counts anti-run compositions, ranks A333489.

A011782 comps counts compositions.

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.

See the formula section for the relationships with A005117, A028234.

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], 1); \\ Amiram Eldar, Oct 26 2024

Formula

{a(n)} = {k >= 1 : A028234(k) is in A005117}. - Peter Munn, May 09 2025