A318680 - OEIS

%I #16 Dec 07 2020 08:55:15

%S 1,2,9,4,25,18,49,8,27,50,121,36,169,98,225,48,289,54,361,100,441,242,

%T 529,72,-125,338,405,196,841,450,961,96,1089,578,1225,108,1369,722,

%U 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 a(n) = n * A318653(n).

%C 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 A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A318653.

%H Antti Karttunen, <a href="/A318680/b318680.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = n * A318653(n).

%F a(n)/A299150(n) = A318511(n)/A318512(n).

%t 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 (PARI)

%o up_to = 65537;

%o A007947(n) = factorback(factorint(n)[, 1]);

%o 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 v318653_aux = DirSqrt(vector(up_to, n, A007947(n)));

%o A318653(n) = numerator(v318653_aux[n]);

%o A318680(n) = (n*A318653(n));

%Y Cf. A007947, A064549, A299150, A318511, A318512, A318653, A318681.

%K sign,mult

%O 1,2

%A _Antti Karttunen_, Sep 02 2018