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A375397 Numbers divisible by the square of some prime factor other than the least. Non-hooklike numbers.

%I A375397 #19 May 08 2025 15:28:15
%S A375397 18,36,50,54,72,75,90,98,100,108,126,144,147,150,162,180,196,198,200,
%T A375397 216,225,234,242,245,250,252,270,288,294,300,306,324,338,342,350,360,
%U A375397 363,375,378,392,396,400,414,432,441,450,468,484,486,490,500,504,507,522
%N A375397 Numbers divisible by the square of some prime factor other than the least. Non-hooklike numbers.
%C A375397 Contains no squarefree numbers A005117 or prime powers A000961, but some perfect powers A131605.
%C A375397 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 A375397 Includes all terms of A036785 = non-products of a squarefree number and a prime power.
%C A375397 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 A375397 Amiram Eldar, <a href="/A375397/b375397.txt">Table of n, a(n) for n = 1..10000</a>
%e A375397 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 A375397 The terms together with their prime indices begin:
%e A375397     18: {1,2,2}
%e A375397     36: {1,1,2,2}
%e A375397     50: {1,3,3}
%e A375397     54: {1,2,2,2}
%e A375397     72: {1,1,1,2,2}
%e A375397     75: {2,3,3}
%e A375397     90: {1,2,2,3}
%e A375397     98: {1,4,4}
%e A375397    100: {1,1,3,3}
%e A375397    108: {1,1,2,2,2}
%e A375397    126: {1,2,2,4}
%e A375397    144: {1,1,1,1,2,2}
%t A375397 Select[Range[100],!SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
%o A375397 (PARI) is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) > e[1], 0); \\ _Amiram Eldar_, Oct 26 2024
%Y A375397 A superset of A036785.
%Y A375397 The complement for maxima is A065200, counted by A034296.
%Y A375397 For maxima instead of minima we have A065201, counted by A239955.
%Y A375397 A version for compositions is A374520, counted by A374640.
%Y A375397 Also positions of non-constant rows in A375128, sums A374706, ranks A375400.
%Y A375397 The complement is A375396, counted by A115029.
%Y A375397 The complement for distinct minima is A375398, counted by A375134.
%Y A375397 For distinct instead of identical minima we have A375399, counts A375404.
%Y A375397 Partitions of this type are counted by A375405.
%Y A375397 A000041 counts integer partitions, strict A000009.
%Y A375397 A003242 counts anti-run compositions, ranks A333489.
%Y A375397 A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
%Y A375397 A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
%Y A375397 Both have length A001222, distinct A001221.
%Y A375397 Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.
%K A375397 nonn
%O A375397 1,1
%A A375397 _Gus Wiseman_, Aug 16 2024
%E A375397 Name edited by _Peter Munn_, May 08 2025