A375397 - OEIS

%I #13 Oct 26 2024 10:45:14

%S 18,36,50,54,72,75,90,98,100,108,126,144,147,150,162,180,196,198,200,

%T 216,225,234,242,245,250,252,270,288,294,300,306,324,338,342,350,360,

%U 363,375,378,392,396,400,414,432,441,450,468,484,486,490,500,504,507,522

%N Numbers divisible by the square of some prime factor except (possibly) the least. Non-hooklike numbers.

%C Contains no squarefree numbers A005117 or prime powers A000961, but some perfect powers A131605.

%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 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.

%C Includes all terms of A036785 = non-products of a squarefree number and a prime power.

%C 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

%H Amiram Eldar, <a href="/A375397/b375397.txt">Table of n, a(n) for n = 1..10000</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 in the sequence.

%e The terms together with their prime indices begin:

%e     18: {1,2,2}

%e     36: {1,1,2,2}

%e     50: {1,3,3}

%e     54: {1,2,2,2}

%e     72: {1,1,1,2,2}

%e     75: {2,3,3}

%e     90: {1,2,2,3}

%e     98: {1,4,4}

%e    100: {1,1,3,3}

%e    108: {1,1,2,2,2}

%e    126: {1,2,2,4}

%e    144: {1,1,1,1,2,2}

%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], 0); \\ _Amiram Eldar_, Oct 26 2024

%Y A superset of A036785.

%Y The complement for maxima is A065200, counted by A034296.

%Y For maxima instead of minima we have A065201, counted by A239955.

%Y A version for compositions is A374520, counted by A374640.

%Y Also positions of non-constant rows in A375128, sums A374706, ranks A375400.

%Y The complement is A375396, counted by A115029.

%Y The complement for distinct minima is A375398, counted by A375134.

%Y For distinct instead of identical minima we have A375399, counts A375404.

%Y Partitions of this type are counted by A375405.

%Y A000041 counts integer partitions, strict A000009.

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

%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 1,1

%A _Gus Wiseman_, Aug 16 2024