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The n-th term of the n-th Dirichlet self-convolution equals n^2.
0

%I #2 Mar 30 2012 18:37:18

%S 1,2,3,-2,5,-24,7,-20,-27,-80,11,-54,13,-168,-195,-394,17,-288,19,

%T -170,-399,-440,23,-9360,-275,-624,-792,-350,29,-2580,31,-72756,-1023,

%U -1088,-1155,-93204,37,-1368,-1443,-87840,41,-8568,43,-902,-10845,-2024,47

%N The n-th term of the n-th Dirichlet self-convolution equals n^2.

%F a(p) = p for prime p.

%e Initial terms of the successive Dirichlet self-convolutions begin:

%e (1),2,3,-2,5,-24,7,-20,-27,-80,11,-54,13,-168,-195,-394,...

%e 1,(4),6,0,10,-36,14,-48,-45,-140,22,-216,26,-308,-360,-864,...

%e 1,6,(9),6,15,-36,21,-76,-54,-180,33,-450,39,-420,-495,-1434,...

%e 1,8,12,(16),20,-24,28,-96,-54,-200,44,-720,52,-504,-600,-2112,...

%e 1,10,15,30,(25),0,35,-100,-45,-200,55,-990,65,-560,-675,-2890,...

%e 1,12,18,48,30,(36),42,-80,-27,-180,66,-1224,78,-588,-720,-3744,...

%e 1,14,21,70,35,84,(49),-28,0,-140,77,-1386,91,-588,-735,-4634,...

%e 1,16,24,96,40,144,56,(64),36,-80,88,-1440,104,-560,-720,-5504,...

%e 1,18,27,126,45,216,63,204,(81),0,99,-1350,117,-504,-675,-6282,...

%e 1,20,30,160,50,300,70,400,135,(100),110,-1080,130,-420,-600,-6880,...

%e 1,22,33,198,55,396,77,660,198,220,(121),-594,143,-308,-495,-7194,...

%e 1,24,36,240,60,504,84,992,270,360,132,(144),156,-168,-360,-7104,...

%e ...

%e where the main diagonal forms the squares.

%o (PARI) {a(n,m)=if(m==0,0^n,if(m==1,A[n],sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

%o A=[1];print1(A[1],",");for(n=2,47,A=concat(A,0);print1(A[n]=n-a(n,n)/n,","))

%K sign

%O 1,2

%A _Paul D. Hanna_, Aug 01 2009