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A318956 For any number n > 0 with prime factorization Product_{k=1..w} p_k ^ x_k (where p_1 < p_2 < ... < p_w and x_k > 0 for k=1..w), let (o_1, ..., o_w) be the ordinal transform of (x_1, ..., x_w); a(n) = Product_{k=1..w} p_k ^ o_k.

%I A318956 #10 Sep 24 2018 08:55:11
%S A318956 1,2,3,2,5,18,7,2,3,50,11,6,13,98,75,2,17,6,19,10,147,242,23,6,5,338,
%T A318956 3,14,29,2250,31,2,363,578,245,18,37,722,507,10,41,6174,43,22,15,1058,
%U A318956 47,6,7,10,867,26,53,6,605,14,1083,1682,59,150,61,1922,21,2
%N A318956 For any number n > 0 with prime factorization Product_{k=1..w} p_k ^ x_k (where p_1 < p_2 < ... < p_w and x_k > 0 for k=1..w), let (o_1, ..., o_w) be the ordinal transform of (x_1, ..., x_w); a(n) = Product_{k=1..w} p_k ^ o_k.
%C A318956 The ordinal transform of a sequence b(n) is the sequence t(n) = number of values in b(1), ..., b(n) which are equal to b(n).
%F A318956 a(p) = p iff p = 1 or p is prime.
%F A318956 a(n^k) = a(n) for any n > 0 and k > 0.
%F A318956 A007947(a(n)) = A007947(n) for any n > 0.
%F A318956 a(a(a(n))) = a(n) for any n > 0.
%F A318956 a(n) belongs to A005117 iff n belongs to A130091 and vice versa.
%F A318956 a(A002110(n)) = A076954(n) for any n >= 0.
%F A318956 a(A076954(n)) = A002110(n) for any n >= 0.
%o A318956 (PARI) a(n) = if (n==1, 1, my (f=factor(n), o=vector(vecmax(f[,2]))); for (i=1, #f~, f[i,2] = o[f[i,2]]++); factorback(f))
%Y A318956 Cf. A002110, A005117, A076954, A130091.
%K A318956 nonn
%O A318956 1,2
%A A318956 _Rémy Sigrist_, Sep 20 2018