|
%I #15 Apr 09 2022 15:11:26
%S 0,0,0,3,4,10,14,24,32,51,56,77,95,128,146,198,225,280,318,384,416,
%T 524,549,672,726,861,915,1063,1143,1292,1382,1551,1661,1867,1966,2211,
%U 2355,2618,2762,3094,3263,3602,3798,4185,4409,4869,5078,5524,5794,6264
%N Sum of the squarefree parts of the partitions of n into 3 parts.
%H David A. Corneth, <a href="/A309478/b309478.txt">Table of n, a(n) for n = 0..9999</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (i * mu(i)^2 + j * mu(j)^2 + (n-i-j) * mu(n-i-j)^2), where mu is the Möbius function (A008683).
%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e 1+1+8
%e 1+1+7 1+2+7
%e 1+2+6 1+3+6
%e 1+1+6 1+3+5 1+4+5
%e 1+1+5 1+2+5 1+4+4 2+2+6
%e 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
%e -----------------------------------------------------------------------
%e n | 3 4 5 6 7 8 9 10 ...
%e -----------------------------------------------------------------------
%e a(n) | 3 4 10 14 24 32 51 56 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[i*MoebiusMu[i]^2 + j*MoebiusMu[j]^2 + (n - i - j) MoebiusMu[n - i - j]^2, {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 50}]
%t Table[Total[Select[Flatten[IntegerPartitions[n,{3}]],SquareFreeQ]],{n,0,50}] (* _Harvey P. Dale_, Apr 09 2022 *)
%Y Cf. A008683, A309455, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.
%K nonn
%O 0,4
%A _Wesley Ivan Hurt_, Aug 04 2019
|
