A163591 The n-th term of the n-th Dirichlet self-convolution equals n^2.

1, 2, 3, -2, 5, -24, 7, -20, -27, -80, 11, -54, 13, -168, -195, -394, 17, -288, 19, -170, -399, -440, 23, -9360, -275, -624, -792, -350, 29, -2580, 31, -72756, -1023, -1088, -1155, -93204, 37, -1368, -1443, -87840, 41, -8568, 43, -902, -10845, -2024, 47

Offset

1,2

Links

Table of n, a(n) for n=1..47.

Formula

a(p) = p for prime p.

Example

Initial terms of the successive Dirichlet self-convolutions begin:
(1),2,3,-2,5,-24,7,-20,-27,-80,11,-54,13,-168,-195,-394,...
1,(4),6,0,10,-36,14,-48,-45,-140,22,-216,26,-308,-360,-864,...
1,6,(9),6,15,-36,21,-76,-54,-180,33,-450,39,-420,-495,-1434,...
1,8,12,(16),20,-24,28,-96,-54,-200,44,-720,52,-504,-600,-2112,...
1,10,15,30,(25),0,35,-100,-45,-200,55,-990,65,-560,-675,-2890,...
1,12,18,48,30,(36),42,-80,-27,-180,66,-1224,78,-588,-720,-3744,...
1,14,21,70,35,84,(49),-28,0,-140,77,-1386,91,-588,-735,-4634,...
1,16,24,96,40,144,56,(64),36,-80,88,-1440,104,-560,-720,-5504,...
1,18,27,126,45,216,63,204,(81),0,99,-1350,117,-504,-675,-6282,...
1,20,30,160,50,300,70,400,135,(100),110,-1080,130,-420,-600,-6880,...
1,22,33,198,55,396,77,660,198,220,(121),-594,143,-308,-495,-7194,...
1,24,36,240,60,504,84,992,270,360,132,(144),156,-168,-360,-7104,...
...
where the main diagonal forms the squares.

Prog

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

Crossrefs

Sequence in context: A084317 A361320 A037279 * A085307 A361581 A361580

Adjacent sequences: A163588 A163589 A163590 * A163592 A163593 A163594

Keyword

sign

Author

Paul D. Hanna, Aug 01 2009

Status

approved