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%I #5 Oct 20 2022 16:27:04
%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T 29,30,31,32,33,34,35,36,37,38,39,41,42,43,44,46,47,49,50,51,52,53,54,
%U 55,56,57,58,59,61,62,64,65,66,67,68,69,70,71,72,73,74
%N Numbers whose prime indices have strictly increasing run-sums. Heinz numbers of the partitions counted by A304428.
%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 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 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 1: {}
%e 2: {1}
%e 3: {2}
%e 4: {1,1}
%e 5: {3}
%e 6: {1,2}
%e 7: {4}
%e 8: {1,1,1}
%e 9: {2,2}
%e 10: {1,3}
%e 11: {5}
%e 13: {6}
%e 14: {1,4}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 17: {7}
%e 18: {1,2,2}
%e 19: {8}
%e For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are not strictly 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],Less@@Total/@Split[primeMS[#]]&]
%Y These partitions are counted by A304428.
%Y The complement is A357863.
%Y These are the indices of rows in A354584 that are strictly increasing.
%Y The opposite (strictly decreasing) version is A357864, counted by A304430.
%Y The weakly increasing version is A357875, counted by A304405.
%Y A001222 counts prime factors, distinct A001221.
%Y A056239 adds up prime indices, row sums of A112798.
%Y Cf. A118914, A181819, A275870, A300273, A304442.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 19 2022
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