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%I #9 May 25 2022 01:22:13
%S 1,2,2,3,2,3,2,4,3,3,2,3,2,3,3,5,2,4,2,4,3,3,2,4,3,3,4,4,2,4,2,6,3,3,
%T 3,4,2,3,3,4,2,4,2,4,4,3,2,5,3,4,3,4,2,5,3,5,3,3,2,4,2,3,3,7,3,4,2,4,
%U 3,4,2,5,2,3,4,4,3,4,2,5,5,3,2,4,3,3,3
%N Number of distinct weak run-sums of the prime indices of n.
%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 A weak run-sum of a sequence is the sum of any consecutive constant subsequence.
%e The prime indices of 72 are {1,1,1,2,2}, with weak runs {}, {1}, {1,1}, {1,1,1}, {2}, {2,2}, which have sums 0, 1, 2, 3, 2, 4, of which 5 are distinct, so a(72) = 5.
%t Table[Length[Union@@Cases[FactorInteger[n],{p_,k_}:>Range[0,k]*PrimePi[p]]],{n,100}]
%Y Positions of 2's are A000040.
%Y Positions of first appearances are A000079.
%Y The strong version is A353835, firsts A002110.
%Y Partitions with distinct run-sums are ranked by A353838, counted by A353837.
%Y The strong version for compositions is A353849.
%Y The greatest run-sum is given by A353862, least A353931.
%Y A001222 counts prime factors, distinct A001221.
%Y A005811 counts runs in binary expansion.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A124010 gives prime signature, sorted A118914.
%Y A165413 counts distinct run-lengths in binary expansion, sums A353929.
%Y A300273 ranks collapsible partitions, counted by A275870.
%Y A353832 represents taking run-sums of a partition, compositions A353847.
%Y A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353840-A353846 pertain to partition run-sum trajectory.
%Y A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353834, A353839, A353866, A353867, A353930.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 23 2022
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