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A325273 Prime omicron of n!.

%I A325273 #14 Apr 18 2021 03:09:59
%S A325273 0,0,1,2,2,2,3,2,2,2,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A325273 2,2,2,2,2,3,3,3,3,3,2,2,2,2,2,2,2,3,3,3,3,2,2,2,3,3,3,3,2,2,2,2,2,2,
%U A325273 2,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A325273 Prime omicron of n!.
%C A325273 We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
%C A325273 The prime omicron of n (A304465) is 0 if n is 1, 1 if n is prime, and otherwise the second-to-last part of the omega-sequence of n. For example, the prime omicron of 180 is 2.
%C A325273 Conjecture: all terms after a(10) = 4 are less than 4.
%C A325273 From _James Rayman_, Apr 17 2021: (Start)
%C A325273 The conjecture is false. a(3804) = 4. In fact, there are 91 values of n < 10000 such that a(n) = 4.
%C A325273 The first value of n such that a(n) = 5 is 37934. For any other n < 5*10^5, a(n) < 5. (End)
%H A325273 James Rayman, <a href="/A325273/b325273.txt">Table of n, a(n) for n = 0..10000</a>
%t A325273 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325273 omicron[n_]:=Switch[n,1,0,_?PrimeQ,1,_,omseq[n][[-2]]];
%t A325273 Table[omicron[n!],{n,0,100}]
%o A325273 (Python)
%o A325273 from sympy.ntheory import *
%o A325273 def red(v):
%o A325273     r = {}
%o A325273     for i in v: r[i] = r.get(i, 0) + 1
%o A325273     return r
%o A325273 def omicron(v):
%o A325273     if len(v) == 0: return 0
%o A325273     if len(v) == 1: return v[0]
%o A325273     else: return omicron(list(red(v).values()))
%o A325273 f, a_list = {}, []
%o A325273 for i in range(101):
%o A325273     a_list.append(omicron(list(f.values())))
%o A325273     g = factorint(i+1)
%o A325273     for k in g: f[k] = f.get(k, 0) + g[k]
%o A325273 print(a_list) # _James Rayman_, Apr 17 2021
%Y A325273 a(n) = A055396(A325275(n)/2).
%Y A325273 Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325274, A325275, A325276, A325277.
%Y A325273 Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
%K A325273 nonn
%O A325273 0,4
%A A325273 _Gus Wiseman_, Apr 18 2019
%E A325273 More terms from _James Rayman_, Apr 17 2021