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A375396
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Numbers not divisible by the square of any prime factor except (possibly) the least. Hooklike numbers.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range[100], SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]], UnsameQ]&]
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CROSSREFS
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The complement is a superset of A036785.
Partitions of this type are counted by A115029.
For distinct instead of identical minima we have A375398, counts A375134.
Cf. A000005, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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