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A353745 Number of runs in the ordered prime signature of n.

%I A353745 #13 Jan 20 2025 16:22:08
%S A353745 0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,1,1,1,
%T A353745 1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,1,2,1,1,1,1,2,
%U A353745 1,1,1,2,1,1,2,2,1,1,1,2,1,1,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,2,2,1,1,1,1,2,1
%N A353745 Number of runs in the ordered prime signature of n.
%C A353745 First differs from A071625 at a(90) = 3.
%C A353745 First differs from A331592 at a(90) = 3.
%C A353745 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%H A353745 Antti Karttunen, <a href="/A353745/b353745.txt">Table of n, a(n) for n = 1..65537</a>
%H A353745 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%H A353745 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A353745 The prime indices of 630 are {1,2,2,3,4}, with multiplicities {1,2,1,1}, with runs {{1},{2},{1,1}}, so a(630) = 3.
%t A353745 Table[Length[Split[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]
%o A353745 (PARI)
%o A353745 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 A353745 runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i < #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
%o A353745 A353745(n) = #runlengths(runlengths(pis_to_runs(n))); \\ _Antti Karttunen_, Jan 20 2025
%Y A353745 Positions of first appearances are A354233.
%Y A353745 A001222 counts prime factors, distinct A001221.
%Y A353745 A005361 gives product of prime signature, firsts A353500/A085629.
%Y A353745 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353745 A124010 gives prime signature, sorted A118914.
%Y A353745 A181819 gives prime shadow, with an inverse A181821.
%Y A353745 A182850/A323014 give frequency depth, counted by A225485/A325280.
%Y A353745 Cf. A005811, A097318, A130091, A304678, A333755, A353503, A353507, A353742.
%Y A353745 Cf. also A329747.
%K A353745 nonn
%O A353745 1,12
%A A353745 _Gus Wiseman_, May 20 2022