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A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

%I A375399 #6 Aug 17 2024 09:02:14
%S A375399 4,8,9,12,16,20,24,25,27,28,32,36,40,44,45,48,49,52,54,56,60,63,64,68,
%T A375399 72,76,80,81,84,88,92,96,99,100,104,108,112,116,117,120,121,124,125,
%U A375399 128,132,135,136,140,144,148,152,153,156,160,162,164,168,169,171
%N A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.
%C A375399 An anti-run is a sequence with no adjacent equal terms.
%C A375399 The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
%C A375399 Note the prime factors can alternatively be taken in weakly decreasing order.
%H A375399 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A375399 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 A375399 The prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is not in the sequence.
%e A375399 The terms together with their prime indices begin:
%e A375399      4: {1,1}
%e A375399      8: {1,1,1}
%e A375399      9: {2,2}
%e A375399     12: {1,1,2}
%e A375399     16: {1,1,1,1}
%e A375399     20: {1,1,3}
%e A375399     24: {1,1,1,2}
%e A375399     25: {3,3}
%e A375399     27: {2,2,2}
%e A375399     28: {1,1,4}
%e A375399     32: {1,1,1,1,1}
%e A375399     36: {1,1,2,2}
%e A375399     40: {1,1,1,3}
%e A375399     44: {1,1,5}
%e A375399     45: {2,2,3}
%e A375399     48: {1,1,1,1,2}
%t A375399 Select[Range[100],!UnsameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
%Y A375399 The complement for compositions is A374638, counted by A374518.
%Y A375399 A version for compositions is A374639, counted by A374678.
%Y A375399 Positions of non-strict rows in A375128, sums A374706, ranks A375400.
%Y A375399 For identical instead of strict we have A375397, counted by A375405.
%Y A375399 The complement is A375398, counted by A375134.
%Y A375399 The complement for maxima instead of minima is A375402, counted by A375133.
%Y A375399 For maxima instead of minima we have A375403, counted by A375401.
%Y A375399 Partitions (or reversed partitions) of this type are counted by A375404.
%Y A375399 A000041 counts integer partitions, strict A000009.
%Y A375399 A003242 counts anti-run compositions, ranks A333489.
%Y A375399 A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
%Y A375399 A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
%Y A375399 Both have length A001222, distinct A001221.
%Y A375399 Cf. A034296, A046660, A065200, A065201, A115029, A279790, A374632, A374761, A375136, A375396.
%K A375399 nonn
%O A375399 1,1
%A A375399 _Gus Wiseman_, Aug 16 2024