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The run-sums of the prime indices of n are not weakly increasing.
6

%I #8 Oct 18 2022 07:27:35

%S 24,45,48,80,90,96,120,135,160,168,175,180,189,192,224,240,264,270,

%T 275,288,297,312,315,320,336,350,360,378,384,405,408,448,456,480,495,

%U 525,528,539,540,550,552,560,567,576,585,594,600,624,630,637,640,672,696

%N The run-sums of the prime indices of n are not 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 terms together with their prime indices begin:

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

%e 45: {2,2,3}

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

%e 80: {1,1,1,1,3}

%e 90: {1,2,2,3}

%e 96: {1,1,1,1,1,2}

%e 120: {1,1,1,2,3}

%e 135: {2,2,2,3}

%e 160: {1,1,1,1,1,3}

%e 168: {1,1,1,2,4}

%e 175: {3,3,4}

%e 180: {1,1,2,2,3}

%e 189: {2,2,2,4}

%e 192: {1,1,1,1,1,1,2}

%e 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 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 are the indices of rows in A354584 that are not weakly increasing.

%Y The complement is A357875.

%Y These partitions are counted by A357878.

%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 1,1

%A _Gus Wiseman_, Oct 17 2022