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%I #7 Aug 17 2024 09:02:22
%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.
%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]&]
%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, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.
%K nonn,new
%O 1,1
%A _Gus Wiseman_, Aug 16 2024
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