cp's OEIS Frontend

A262351 Sum of the parts in the partitions of n into exactly two squarefree parts.

%I A262351 #23 Aug 19 2021 12:26:53
%S A262351 0,2,3,8,5,12,14,24,18,20,22,48,39,42,45,80,68,72,57,120,84,110,92,
%T A262351 168,125,130,135,196,145,150,155,256,198,238,210,396,259,266,273,440,
%U A262351 328,336,387,572,450,368,376,624,490,400,357,728,530,540,385,728,570
%N A262351 Sum of the parts in the partitions of n into exactly two squarefree parts.
%C A262351 One-half of the sum of the perimeters of the rectangles with squarefree length and width such that L + W = n, W <= L. For example, a(8) = 24; the rectangles are 1 X 7, 2 X 6 and 3 X 5 with perimeters 16, 16 and 16. Then (16 + 16 + 16)/2 = 48/2 = 24. - _Wesley Ivan Hurt_, Nov 04 2017
%H A262351 Peter Kagey, <a href="/A262351/b262351.txt">Table of n, a(n) for n = 1..10000</a>
%H A262351 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A262351 a(n) = n * Sum_{i=1..floor(n/2)} mu(i)^2 * mu(n-i)^2, where mu is the Möbius function (A008683).
%F A262351 a(n) = n * A071068(n).
%e A262351 a(4) = 8; There are two partitions of 4 into two squarefree parts: (3,1) and (2,2). Thus we have a(4) = (3+1) + (2+2) = 8.
%e A262351 a(7) = 14; There are three partitions of 7 into two parts: (6,1), (5,2) and (4,3). Since only two of these partitions have squarefree parts, we have a(7) = (6+1) + (5+2) = 14.
%t A262351 Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]
%t A262351 Table[Total[Flatten[Select[IntegerPartitions[n,{2}],AllTrue[ #,SquareFreeQ]&]]],{n,60}] (* _Harvey P. Dale_, Aug 19 2021 *)
%o A262351 (PARI) a(n)=my(s=issquarefree(n-1) && n>1); forfactored(k=(n+1)\2,n-2, if(vecmax(k[2][,2])==1 && issquarefree(n-k[1]), s++)); s*n \\ _Charles R Greathouse IV_, Nov 05 2017
%Y A262351 Cf. A005117, A008683, A071068.
%K A262351 nonn,easy
%O A262351 1,2
%A A262351 _Wesley Ivan Hurt_, Sep 18 2015