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A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

%I A385576 #8 Jul 05 2025 09:59:39
%S A385576 1,2,3,5,7,11,12,13,17,18,19,20,23,28,29,31,37,41,43,44,45,47,50,52,
%T A385576 53,59,61,63,67,68,71,73,75,76,79,83,89,92,97,98,99,101,103,107,109,
%U A385576 113,116,117,120,124,127,131,137,139,147,148,149,151,153,157,163
%N A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.
%C A385576 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A385576 These are also numbers with the same number of adjacent equal prime indices as adjacent unequal prime indices.
%F A385576 A001221(a(n)) = A375136(a(n)).
%e A385576 The prime indices of 2640 are {1,1,1,1,2,3,5}, with 4 distinct parts {1,2,3,5} and 4 maximal anti-runs ((1),(1),(1),(2,3,5)), so 2640 is in the sequence.
%e A385576 The terms together with their prime indices begin:
%e A385576    1: {}
%e A385576    2: {1}
%e A385576    3: {2}
%e A385576    5: {3}
%e A385576    7: {4}
%e A385576   11: {5}
%e A385576   12: {1,1,2}
%e A385576   13: {6}
%e A385576   17: {7}
%e A385576   18: {1,2,2}
%e A385576   19: {8}
%e A385576   20: {1,1,3}
%e A385576   23: {9}
%e A385576   28: {1,1,4}
%e A385576   29: {10}
%e A385576   31: {11}
%e A385576   37: {12}
%e A385576   41: {13}
%e A385576   43: {14}
%e A385576   44: {1,1,5}
%e A385576   45: {2,2,3}
%e A385576   47: {15}
%t A385576 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A385576 Select[Range[100],#==1||PrimeNu[#]==Length[Split[prix[#],UnsameQ]]&]
%Y A385576 The LHS is the rank statistic A001221, triangle counted by A116608.
%Y A385576 The RHS is the rank statistic A375136, triangle counted by A133121.
%Y A385576 These partitions are counted by A385574.
%Y A385576 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A385576 A047993 counts partitions with max part = length, ranks A106529.
%Y A385576 A356235 counts partitions with a neighborless singleton, ranks A356237.
%Y A385576 A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
%Y A385576 A384893 counts subsets by maximal anti-runs, for partitions A268193, strict A384905.
%Y A385576 A385572 counts subsets with the same number of runs as anti-runs, ranks A385575.
%Y A385576 Cf. A044813, A046660, A210034, A297155, A356226, A361205, A384889, A385213.
%K A385576 nonn
%O A385576 1,2
%A A385576 _Gus Wiseman_, Jul 04 2025