%I #4 Oct 18 2022 13:32:32
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,
%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,46,47,49,50,51,52,
%U 53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N Numbers whose run-sums of prime indices are weakly increasing.
%C 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 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 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e The prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is not in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],LessEqual@@Total/@Split[primeMS[#]]&]
%Y These partitions are counted by A304405.
%Y These are the indices of rows in A354584 that are weakly increasing.
%Y The complement is A357876.
%Y A001222 counts prime factors, distinct A001221.
%Y A056239 adds up prime indices, row sums of A112798.
%Y Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 18 2022