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%I A357864 #7 Oct 20 2022 13:12:07
%S A357864 1,2,3,4,5,7,8,9,11,13,16,17,19,23,24,25,27,29,31,32,37,41,43,45,47,
%T A357864 48,49,53,59,61,64,67,71,73,79,80,81,83,89,96,97,101,103,107,109,113,
%U A357864 121,125,127,128,131,135,137,139,149,151,157,160,163,167,169,173
%N A357864 Numbers whose prime indices have strictly decreasing run-sums. Heinz numbers of the partitions counted by A304430.
%C A357864 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 A357864 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 A357864 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 A357864 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357864 The terms together with their prime indices begin:
%e A357864     1: {}
%e A357864     2: {1}
%e A357864     3: {2}
%e A357864     4: {1,1}
%e A357864     5: {3}
%e A357864     7: {4}
%e A357864     8: {1,1,1}
%e A357864     9: {2,2}
%e A357864    11: {5}
%e A357864    13: {6}
%e A357864    16: {1,1,1,1}
%e A357864    17: {7}
%e A357864    19: {8}
%e A357864    23: {9}
%e A357864    24: {1,1,1,2}
%e A357864    25: {3,3}
%e A357864    27: {2,2,2}
%e A357864    29: {10}
%e A357864 For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are strictly decreasing, so 24 is in the sequence.
%t A357864 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357864 Select[Range[300],Greater@@Total/@Split[primeMS[#]]&]
%Y A357864 Subsequence of A304686.
%Y A357864 These partitions are counted by A304430.
%Y A357864 These are the indices  of rows in A354584 that are strictly decreasing.
%Y A357864 The weakly decreasing version is A357861, counted by A304406.
%Y A357864 The opposite version is A357862, counted by A304428, complement A357863.
%Y A357864 A001222 counts prime factors, distinct A001221.
%Y A357864 A056239 adds up prime indices, row sums of A112798.
%Y A357864 Cf. A118914, A181819, A300273, A304405, A304442, A357875.
%K A357864 nonn
%O A357864 1,2
%A A357864 _Gus Wiseman_, Oct 19 2022

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