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A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.

%I A323022 #16 Feb 11 2019 19:10:43
%S A323022 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 A323022 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 A323022 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 A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.
%C A323022 The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}.
%C A323022 First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k.
%C A323022 The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n.
%C A323022 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 A323022 Antti Karttunen, <a href="/A323022/b323022.txt">Table of n, a(n) for n = 1..100000</a>
%F A323022 a(n) = A071625(A181819(n))
%F A323022      = A001221(A181819(A181819(n)))
%F A323022      = A001222(A181819(A181819(A181819(n))))
%F A323022      = A056239(A181819(A181819(A181819(A181819(n))))).
%e A323022 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 A323022 red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
%t A323022 Table[PrimeNu[red[red[n]]],{n,200}]
%o A323022 (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 A323022 (PARI)
%o A323022 A071625(n) = #Set(factor(n)[, 2]); \\ From A071625
%o A323022 A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A323022 A323022(n) = A071625(A181819(n)); \\ _Antti Karttunen_, Jan 03 2019
%Y A323022 Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023.
%K A323022 nonn
%O A323022 1,60
%A A323022 _Gus Wiseman_, Jan 02 2019
%E A323022 More terms from _Antti Karttunen_, Jan 03 2019