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%I #16 Feb 11 2019 19:10:43
%S 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Fourth omega of n. Number of distinct multiplicities in the prime signature of n.
%C The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}.
%C First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k.
%C The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n.
%C We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first three omegas are A001222, A001221, A071625, and this sequence is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices).
%H Antti Karttunen, <a href="/A323022/b323022.txt">Table of n, a(n) for n = 1..100000</a>
%F a(n) = A071625(A181819(n))
%F = A001221(A181819(A181819(n)))
%F = A001222(A181819(A181819(A181819(n))))
%F = A056239(A181819(A181819(A181819(A181819(n))))).
%e The prime signature of 1286485200 is {1, 1, 1, 2, 2, 3, 4}, in which 1 appears three times, two appears twice, and 3 and 4 both appear once, so there are 3 distinct multiplicities {1, 2, 3} and hence a(1286485200) = 3.
%t red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
%t Table[PrimeNu[red[red[n]]],{n,200}]
%o (PARI) a(n) = my(e=factor(n)[, 2], s = Set(e), m=Map(), v=vector(#s)); for(i=1, #s, mapput(m,s[i],i)); for(i=1, #e, v[mapget(m,e[i])]++); #Set(v) \\ _David A. Corneth_, Jan 02 2019
%o (PARI)
%o A071625(n) = #Set(factor(n)[, 2]); \\ From A071625
%o A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A323022(n) = A071625(A181819(n)); \\ _Antti Karttunen_, Jan 03 2019
%Y Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023.
%K nonn
%O 1,60
%A _Gus Wiseman_, Jan 02 2019
%E More terms from _Antti Karttunen_, Jan 03 2019
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