A318681 - OEIS

%I #13 Sep 02 2018 17:07:31

%S 1,2,9,12,25,18,49,40,243,50,121,108,169,98,225,560,289,486,361,300,

%T 441,242,529,360,1875,338,3645,588,841,450,961,2016,1089,578,1225,

%U 2916,1369,722,1521,1000,1681,882,1849,1452,6075,1058,2209,5040,7203,3750,2601,2028,2809,7290,3025,1960,3249,1682,3481,2700

%N a(n) = n * A299149(n).

%C Dirichlet convolution of a(n)/A299150(n) with itself gives A000290, the squares, like gives also the self-convolution of A318649(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318649, this sequence is multiplicative as both A000027 and A299149 are multiplicative sequences (also, because A000290 and A299150 are both multiplicative).

%C A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A299149.

%H Antti Karttunen, <a href="/A318681/b318681.txt">Table of n, a(n) for n = 1..16384</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>

%F a(n) = n * A299149(n).

%F a(n)/A299150(n) = A318649(n)/A318512(n).

%o (PARI)

%o up_to = 65537;

%o DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] -

%o sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};

%o v299149_50 = DirSqrt(vector(up_to, n, n));

%o A299149(n) = numerator(v299149_50[n]);

%o A318681(n) = (n*A299149(n));

%Y Cf. A000290, A299149, A299150, A318512, A318649, A318680.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Sep 02 2018