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A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

%I A357850 #5 Oct 20 2022 12:44:11
%S A357850 6,10,14,15,18,20,21,22,26,28,30,33,34,35,36,38,39,42,44,46,50,51,52,
%T A357850 54,55,56,57,58,60,62,65,66,68,69,70,72,74,75,76,77,78,82,84,85,86,87,
%U A357850 88,90,91,92,93,94,95,98,99,100,102,104,105,106,108,110,111
%N A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.
%C A357850 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 A357850 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A357850 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 A357850 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357850 The terms together with their prime indices begin:
%e A357850     6: {1,2}
%e A357850    10: {1,3}
%e A357850    14: {1,4}
%e A357850    15: {2,3}
%e A357850    18: {1,2,2}
%e A357850    20: {1,1,3}
%e A357850    21: {2,4}
%e A357850    22: {1,5}
%e A357850    26: {1,6}
%e A357850    28: {1,1,4}
%e A357850    30: {1,2,3}
%e A357850    33: {2,5}
%e A357850    34: {1,7}
%e A357850    35: {3,4}
%e A357850    36: {1,1,2,2}
%e A357850    38: {1,8}
%e A357850    39: {2,6}
%e A357850    42: {1,2,4}
%t A357850 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357850 Select[Range[100],!GreaterEqual@@Total/@Split[primeMS[#]]&]
%Y A357850 These are the indices of rows in A354584 that are not weakly decreasing.
%Y A357850 The complement is A357861, counted by A304406.
%Y A357850 These partitions are counted by A357865.
%Y A357850 The opposite (not weakly increasing) version is A357876, counted by A357878.
%Y A357850 A001222 counts prime factors, distinct A001221.
%Y A357850 A056239 adds up prime indices, row sums of A112798.
%Y A357850 Cf. A118914, A181819, A300273, A304405, A304428, A304430, A304442, A353832, A353864, A353932, A357864, A357875.
%K A357850 nonn
%O A357850 1,1
%A A357850 _Gus Wiseman_, Oct 19 2022