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A353861 Number of distinct weak run-sums of the prime indices of n.

%I A353861 #14 Jan 20 2025 22:52:15
%S A353861 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 A353861 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 A353861 3,4,2,5,2,3,4,4,3,4,2,5,5,3,2,4,3,3,3,5,2,5,3,4,3,3,3,6,2,4,4,5,2,4,2,5,4,3,2,5
%N A353861 Number of distinct weak run-sums of the prime indices of n.
%C A353861 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 A353861 A weak run-sum of a sequence is the sum of any consecutive constant subsequence.
%H A353861 Antti Karttunen, <a href="/A353861/b353861.txt">Table of n, a(n) for n = 1..65537</a>
%H A353861 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A353861 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 A353861 Table[Length[Union@@Cases[FactorInteger[n],{p_,k_}:>Range[0,k]*PrimePi[p]]],{n,100}]
%o A353861 (PARI)
%o A353861 pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
%o A353861 A353861(n) = if(1==n,n,my(pruns = pis_to_runs(n), runsum = 0, runsums = List([])); for(i=1,#pruns, listput(runsums, runsum); if((i>1) && pruns[i] == pruns[i-1], runsum += pruns[i], runsum = pruns[i])); listput(runsums, runsum); #Set(runsums)); \\ _Antti Karttunen_, Jan 20 2025
%Y A353861 Positions of 2's are A000040.
%Y A353861 Positions of first appearances are A000079.
%Y A353861 The strong version is A353835, firsts A002110.
%Y A353861 Partitions with distinct run-sums are ranked by A353838, counted by A353837.
%Y A353861 The strong version for compositions is A353849.
%Y A353861 The greatest run-sum is given by A353862, least A353931.
%Y A353861 A001222 counts prime factors, distinct A001221.
%Y A353861 A005811 counts runs in binary expansion.
%Y A353861 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353861 A124010 gives prime signature, sorted A118914.
%Y A353861 A165413 counts distinct run-lengths in binary expansion, sums A353929.
%Y A353861 A300273 ranks collapsible partitions, counted by A275870.
%Y A353861 A353832 represents taking run-sums of a partition, compositions A353847.
%Y A353861 A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353861 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353861 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A353861 Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353834, A353839, A353866, A353867, A353930.
%K A353861 nonn
%O A353861 1,2
%A A353861 _Gus Wiseman_, May 23 2022
%E A353861 Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025