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Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.
6

%I #6 Aug 17 2024 21:50:42

%S 4,8,9,16,18,24,25,27,32,36,40,48,49,50,54,56,64,72,75,80,81,88,96,98,

%T 100,104,108,112,120,121,125,128,135,136,144,147,150,152,160,162,168,

%U 169,176,184,189,192,196,200,208,216,224,225,232,240,242,243,245

%N Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.

%C First differs from A299117 in having 150.

%C An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.

%C The partitions with these Heinz numbers are those with (1) some part appearing more than twice or (2) the greatest part appearing more than once.

%C Note the prime factors can alternatively be written in weakly decreasing order.

%e The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is in the sequence.

%e The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is not in the sequence.

%e The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is in the sequence.

%e The terms together with their prime indices begin:

%e 4: {1,1}

%e 8: {1,1,1}

%e 9: {2,2}

%e 16: {1,1,1,1}

%e 18: {1,2,2}

%e 24: {1,1,1,2}

%e 25: {3,3}

%e 27: {2,2,2}

%e 32: {1,1,1,1,1}

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

%e 40: {1,1,1,3}

%e 48: {1,1,1,1,2}

%t Select[Range[150],!UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

%Y For identical instead of distinct we have A065201, complement A065200.

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

%Y For minima instead of maxima we have A375399, counted by A375404.

%Y Partitions of this type are counted by A375401.

%Y The complement is A375402, counted by A375133.

%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. A046660, A066328, A358836, A374632, A374706, A374768, A374767, A375128, A375136, A375396, A375400.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 15 2024