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A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

%I A304768 #13 Feb 12 2023 01:46:56
%S A304768 1,2,3,2,5,6,7,2,3,10,11,18,13,14,15,2,17,12,19,50,21,22,23,54,5,26,3,
%T A304768 98,29,30,31,2,33,34,35,6,37,38,39,250,41,42,43,242,75,46,47,162,7,20,
%U A304768 51,338,53,24,55,686,57,58,59,450,61,62,147,2,65,66,67
%N A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.
%C A304768 Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.
%H A304768 Amiram Eldar, <a href="/A304768/b304768.txt">Table of n, a(n) for n = 1..10000</a>
%F A304768 If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).
%t A304768 acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
%t A304768 Array[acj,100]
%o A304768 (PARI) a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ _Amiram Eldar_, Feb 12 2023
%Y A304768 Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.
%K A304768 nonn
%O A304768 1,2
%A A304768 _Gus Wiseman_, May 18 2018