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A318680 a(n) = n * A318653(n).

%I A318680 #19 Mar 17 2025 05:37:03
%S A318680 1,2,9,4,25,18,49,8,27,50,121,36,169,98,225,48,289,54,361,100,441,242,
%T A318680 529,72,-125,338,405,196,841,450,961,96,1089,578,1225,108,1369,722,
%U A318680 1521,200,1681,882,1849,484,675,1058,2209,432,-1029,-250,2601,676,2809,810,3025,392,3249,1682,3481,900,3721,1922,1323,320
%N A318680 a(n) = n * A318653(n).
%C A318680 Dirichlet convolution of a(n)/A299150(n) with itself gives A064549 [= n * Product_{primes p|n} p], like gives also the self-convolution of A318511(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318511, this sequence is multiplicative as both A000027 and A318653 are multiplicative sequences (also, because A064549 and A299150 are both multiplicative).
%C A318680 A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A318653.
%H A318680 Antti Karttunen, <a href="/A318680/b318680.txt">Table of n, a(n) for n = 1..65537</a>
%F A318680 a(n) = n * A318653(n).
%F A318680 a(n)/A299150(n) = A318511(n)/A318512(n).
%t A318680 rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := n * Numerator [f[n]]; Array[a, 100] (* _Amiram Eldar_, Dec 07 2020 *)
%o A318680 (PARI)
%o A318680 up_to = 65537;
%o A318680 A007947(n) = factorback(factorint(n)[, 1]);
%o A318680 DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};
%o A318680 v318653_aux = DirSqrt(vector(up_to, n, A007947(n)));
%o A318680 A318653(n) = numerator(v318653_aux[n]);
%o A318680 A318680(n) = (n*A318653(n));
%Y A318680 Cf. A007947, A064549, A299150, A318511, A318512, A318653, A318681.
%K A318680 sign,mult
%O A318680 1,2
%A A318680 _Antti Karttunen_, Sep 02 2018