cp's OEIS Frontend

A325248 Heinz number of the omega-sequence of n.

%I A325248 #8 Aug 22 2019 09:52:23
%S A325248 1,2,2,6,2,18,2,10,6,18,2,90,2,18,18,14,2,90,2,90,18,18,2,126,6,18,10,
%T A325248 90,2,50,2,22,18,18,18,42,2,18,18,126,2,50,2,90,90,18,2,198,6,90,18,
%U A325248 90,2,126,18,126,18,18,2,630,2,18,90,26,18,50,2,90,18,50
%N A325248 Heinz number of the omega-sequence of n.
%C A325248 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 A325248 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A325248 A001222(a(n)) = A323014(n).
%F A325248 A061395(a(n)) = A001222(n).
%F A325248 A304465(n) = A055396(a(n)/2).
%F A325248 A325249(n) = A056239(a(n)).
%F A325248 a(n!) = A325275(n).
%e A325248 The omega-sequence of 180 is (5,3,2,2,1) with Heinz number 990, so a(180) = 990.
%t A325248 omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
%t A325248 Table[Times@@Prime/@omseq[n],{n,100}]
%Y A325248 Positions of squarefree terms are A325247.
%Y A325248 Positions of normal numbers (A055932) are A325251.
%Y A325248 First positions of each distinct term are A325238.
%Y A325248 Cf. A056239, A070175, A112798, A118914, A181819, A181821, A323023, A325249, A325275, A325277.
%Y A325248 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).
%K A325248 nonn
%O A325248 1,2
%A A325248 _Gus Wiseman_, Apr 16 2019