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A357876 The run-sums of the prime indices of n are not weakly increasing.

%I A357876 #8 Oct 18 2022 07:27:35
%S A357876 24,45,48,80,90,96,120,135,160,168,175,180,189,192,224,240,264,270,
%T A357876 275,288,297,312,315,320,336,350,360,378,384,405,408,448,456,480,495,
%U A357876 525,528,539,540,550,552,560,567,576,585,594,600,624,630,637,640,672,696
%N A357876 The run-sums of the prime indices of n are not weakly increasing.
%C A357876 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 A357876 The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
%H A357876 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357876 The terms together with their prime indices begin:
%e A357876    24: {1,1,1,2}
%e A357876    45: {2,2,3}
%e A357876    48: {1,1,1,1,2}
%e A357876    80: {1,1,1,1,3}
%e A357876    90: {1,2,2,3}
%e A357876    96: {1,1,1,1,1,2}
%e A357876   120: {1,1,1,2,3}
%e A357876   135: {2,2,2,3}
%e A357876   160: {1,1,1,1,1,3}
%e A357876   168: {1,1,1,2,4}
%e A357876   175: {3,3,4}
%e A357876   180: {1,1,2,2,3}
%e A357876   189: {2,2,2,4}
%e A357876   192: {1,1,1,1,1,1,2}
%e A357876 For example, the prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is in the sequence.
%t A357876 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357876 Select[Range[100],!LessEqual@@Total/@Split[primeMS[#]]&]
%Y A357876 These are the indices  of rows in A354584 that are not weakly increasing.
%Y A357876 The complement is A357875.
%Y A357876 These partitions are counted by A357878.
%Y A357876 A001222 counts prime factors, distinct A001221.
%Y A357876 A056239 adds up prime indices, row sums of A112798.
%Y A357876 Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.
%K A357876 nonn
%O A357876 1,1
%A A357876 _Gus Wiseman_, Oct 17 2022