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A353835 Number of distinct run-sums of the prime indices of n.

%I A353835 #11 Jan 20 2025 16:22:28
%S A353835 0,1,1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,3,1,1,2,2,
%T A353835 2,2,1,2,2,1,1,3,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,2,1,2,1,1,2,3,1,2,
%U A353835 2,3,1,2,1,2,2,2,2,3,1,2,1,2,1,2,2,2,2,2,1,3,2,2,2,2,2,2,1,2,2,2,1,3,1,2,3
%N A353835 Number of distinct run-sums of the prime indices of n.
%C A353835 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 A353835 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 A353835 Antti Karttunen, <a href="/A353835/b353835.txt">Table of n, a(n) for n = 1..65537</a>
%H A353835 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%H A353835 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A353835 a(n) = A001221(A353832(n)). [From formula section of A353832] -  _Antti Karttunen_, Jan 20 2025
%e A353835 The prime indices of 3780 are {1,1,2,2,2,3,4}, with distinct run-sums {2,3,4,6}, so a(3780) = 4.
%e A353835 The prime indices of 8820 are {1,1,2,2,3,4,4}, with distinct run-sums {2,3,4,8}, so a(8820) = 4.
%e A353835 The prime indices of 13860 are {1,1,2,2,3,4,5}, with distinct run-sums {2,3,4,5}, so a(13860) = 4.
%e A353835 The prime indices of 92400 are {1,1,1,1,2,3,3,4,5}, with distinct run-sums {2,4,5,6}, so a(92400) = 4.
%t A353835 Table[Length[Union[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k]]],{n,100}]
%o A353835 (PARI)
%o A353835 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 A353835 A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
%o A353835 A353835(n) = omega(A353832(n)); \\ _Antti Karttunen_, Jan 20 2025
%Y A353835 Positions of first appearances are A002110.
%Y A353835 A version for binary expansion is A165413.
%Y A353835 Positions of 0's and 1's are A353833, nonprime A353834, counted by A304442.
%Y A353835 The case of all distinct run-sums is ranked by A353838, counted by A353837.
%Y A353835 The version for compositions is A353849.
%Y A353835 The weak version is A353861.
%Y A353835 A001222 counts prime factors, distinct A001221.
%Y A353835 A005811 counts runs in binary expansion.
%Y A353835 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353835 A124010 gives prime signature, sorted A118914.
%Y A353835 A300273 ranks collapsible partitions, counted by A275870.
%Y A353835 A351014 counts distinct runs in standard compositions.
%Y A353835 A353832 represents the operation of taking run-sums of a partition.
%Y A353835 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353835 A353862 gives greatest run-sum of prime indices, least A353931.
%Y A353835 A353866 ranks rucksack partitions, counted by A353864.
%Y A353835 Cf. A071625, A073093, A116608, A175413, A181819, A333755, A353839, A353848, A353850, A353852, A353867.
%K A353835 nonn
%O A353835 1,6
%A A353835 _Gus Wiseman_, May 23 2022
%E A353835 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025