cp's OEIS Frontend

A375403 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.

%I A375403 #6 Aug 17 2024 21:50:42
%S A375403 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 A375403 100,104,108,112,120,121,125,128,135,136,144,147,150,152,160,162,168,
%U A375403 169,176,184,189,192,196,200,208,216,224,225,232,240,242,243,245
%N A375403 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.
%C A375403 First differs from A299117 in having 150.
%C A375403 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 A375403 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 A375403 Note the prime factors can alternatively be written in weakly decreasing order.
%e A375403 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 A375403 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 A375403 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 A375403 The terms together with their prime indices begin:
%e A375403     4: {1,1}
%e A375403     8: {1,1,1}
%e A375403     9: {2,2}
%e A375403    16: {1,1,1,1}
%e A375403    18: {1,2,2}
%e A375403    24: {1,1,1,2}
%e A375403    25: {3,3}
%e A375403    27: {2,2,2}
%e A375403    32: {1,1,1,1,1}
%e A375403    36: {1,1,2,2}
%e A375403    40: {1,1,1,3}
%e A375403    48: {1,1,1,1,2}
%t A375403 Select[Range[150],!UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
%Y A375403 For identical instead of distinct we have A065201, complement A065200.
%Y A375403 The complement for minima is A375398, counted by A375134.
%Y A375403 For minima instead of maxima we have A375399, counted by A375404.
%Y A375403 Partitions of this type are counted by A375401.
%Y A375403 The complement is A375402, counted by A375133.
%Y A375403 A000041 counts integer partitions, strict A000009.
%Y A375403 A003242 counts anti-run compositions, ranks A333489.
%Y A375403 A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
%Y A375403 A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
%Y A375403 Both have length A001222, distinct A001221.
%Y A375403 Cf. A046660, A066328, A358836, A374632, A374706, A374768, A374767, A375128, A375136, A375396, A375400.
%K A375403 nonn
%O A375403 1,1
%A A375403 _Gus Wiseman_, Aug 15 2024