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%I A357861 #9 Oct 20 2022 12:44:21
%S A357861 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,24,25,27,29,31,32,37,40,41,43,
%T A357861 45,47,48,49,53,59,61,63,64,67,71,73,79,80,81,83,89,96,97,101,103,107,
%U A357861 109,112,113,121,125,127,128,131,135,137,139,144,149,151,157
%N A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.
%C A357861 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 A357861 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 A357861 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 A357861 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357861 The terms together with their prime indices begin:
%e A357861     1: {}
%e A357861     2: {1}
%e A357861     3: {2}
%e A357861     4: {1,1}
%e A357861     5: {3}
%e A357861     7: {4}
%e A357861     8: {1,1,1}
%e A357861     9: {2,2}
%e A357861    11: {5}
%e A357861    12: {1,1,2}
%e A357861    13: {6}
%e A357861    16: {1,1,1,1}
%e A357861    17: {7}
%e A357861    19: {8}
%e A357861    23: {9}
%e A357861    24: {1,1,1,2}
%e A357861    25: {3,3}
%e A357861    27: {2,2,2}
%e A357861 For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.
%t A357861 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357861 Select[Range[100],GreaterEqual@@Total/@Split[primeMS[#]]&]
%Y A357861 These partitions are counted by A304406.
%Y A357861 These are the indices  of rows in A354584 that are weakly decreasing.
%Y A357861 The complement is A357850, counted by A357865, opposite A357876.
%Y A357861 The strictly decreasing version is A357864, counted by A304430.
%Y A357861 The opposite (weakly increasing) version is A357875, counted by A304405.
%Y A357861 A001222 counts prime factors, distinct A001221.
%Y A357861 A056239 adds up prime indices, row sums of A112798.
%Y A357861 Cf. A047966, A118914, A181819, A239312, A300273, A304442.
%K A357861 nonn
%O A357861 1,2
%A A357861 _Gus Wiseman_, Oct 19 2022

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