A375396 - OEIS

%I #18 Oct 26 2024 10:45:19

%S 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,

%T 28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,

%U 53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71

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

%C 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.

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

%C 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

%H Amiram Eldar, <a href="/A375396/b375396.txt">Table of n, a(n) for n = 1..10000</a>

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e 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.

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

%o (PARI) is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) == e[1], 1); \\ _Amiram Eldar_, Oct 26 2024

%Y The complement is a superset of A036785.

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

%Y The complement for maxima is A065201, counted by A239955.

%Y Partitions of this type are counted by A115029.

%Y A version for compositions is A374519, counted by A374517.

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

%Y The complement is A375397, counted by A375405.

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

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

%Y A000041 counts integer partitions, strict A000009.

%Y A003242 counts anti-run compositions, ranks A333489.

%Y A011782 comps counts compositions.

%Y A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.

%Y A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.

%Y Both have length A001222, distinct A001221.

%Y Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 16 2024