cp's OEIS Frontend

A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.

%I A325276 #7 Apr 19 2019 08:10:07
%S A325276 1,2,2,1,4,2,2,1,5,3,2,2,1,7,3,3,1,8,4,3,2,2,1,11,4,3,2,2,1,13,4,3,2,
%T A325276 2,1,15,4,4,1,16,5,4,2,2,1,19,5,4,2,2,1,20,6,4,2,2,1,22,6,4,2,1,24,6,
%U A325276 5,2,2,1,28,6,5,2,2,1,29,7,5,2,2,1
%N A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.
%C A325276 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).
%e A325276 Triangle begins:
%e A325276   {}
%e A325276   {}
%e A325276    1
%e A325276    2  2  1
%e A325276    4  2  2  1
%e A325276    5  3  2  2  1
%e A325276    7  3  3  1
%e A325276    8  4  3  2  2  1
%e A325276   11  4  3  2  2  1
%e A325276   13  4  3  2  2  1
%e A325276   15  4  4  1
%e A325276   16  5  4  2  2  1
%e A325276   19  5  4  2  2  1
%e A325276   20  6  4  2  2  1
%e A325276   22  6  4  2  1
%e A325276   24  6  5  2  2  1
%e A325276   28  6  5  2  2  1
%e A325276   29  7  5  2  2  1
%e A325276   32  7  5  2  2  1
%e A325276   33  8  5  2  2  1
%e A325276   36  8  5  2  2  1
%e A325276   38  8  5  2  2  1
%e A325276   40  8  6  2  2  1
%e A325276   41  9  6  2  2  1
%e A325276   45  9  6  2  2  1
%e A325276   47  9  6  2  2  1
%e A325276   49  9  6  3  2  2  1
%e A325276   52  9  6  3  2  2  1
%e A325276   55  9  6  3  2  2  1
%e A325276   56 10  6  3  2  2  1
%e A325276   59 10  6  3  2  2  1
%t A325276 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325276 Table[omseq[n!],{n,0,30}]
%Y A325276 Row lengths are A325272. Row sums are A325274. Row n is row A325275(n) of A112798. Second-to-last column is A325273. Column k = 1 is A022559. Column k = 2 is A000720. Column k = 3 is A071626.
%Y A325276 Cf. A000142, A006939, A081401, A303555, A323023, A325238, A325275, A325277.
%Y A325276 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 A325276 nonn,tabf
%O A325276 0,2
%A A325276 _Gus Wiseman_, Apr 18 2019