A302391 - OEIS

%I #20 Dec 11 2023 21:45:53

%S 0,0,1,1,3,2,4,3,5,4,6,5,8,7,10,8,10,7,12,9,13,9,15,11,17,12,17,15,18,

%T 15,20,17,22,16,22,15,23,18,26,20,28,20,29,21,28,23,30,24,32,25,33,26,

%U 34,23,36,27,37,27,39,29,41,29,40,34,42,34,44,36,46

%N Number of partitions of 2n into two parts with at least one nonsquarefree part.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..n} 1 - mu(2n-i)^2 * mu(i)^2, where mu is the Möbius function (A008683).

%F a(n) = n - A280226(n). - _Wesley Ivan Hurt_, Dec 11 2023

%t Table[Sum[1 - MoebiusMu[2 n - i]^2*MoebiusMu[i]^2, {i, n}], {n, 100}]

%o (Magma) [&+[(1-MoebiusMu(2*n-k)^2*MoebiusMu(k)^2): k in [1..n]]: n in [1..70]]; // _Vincenzo Librandi_, Apr 09 2018

%o (PARI) a(n) = sum(i=1, n, 1 - moebius(2*n-i)^2*moebius(i)^2); \\ _Michel Marcus_, Apr 09 2018

%Y Cf. A008683, A013929, A280226, A294097.

%K nonn

%O 1,5

%A _Wesley Ivan Hurt_, Apr 06 2018