cp's OEIS Frontend

A294062 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part squarefree.

%I A294062 #23 Jul 08 2025 07:57:23
%S A294062 0,2,6,12,18,26,36,48,60,72,86,102,118,136,156,178,200,224,248,274,
%T A294062 300,328,358,390,422,454,488,522,556,592,630,670,710,752,796,842,888,
%U A294062 936,986,1038,1090,1144,1200,1258,1316,1374,1434,1496,1558,1620,1682,1746
%N A294062 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part squarefree.
%C A294062 Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 at squarefree values of x for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(6), the squarefree values of x are x=1,2,3,5,6 and so a(6) = 12-2*1 + 12-2*2 + 12-2*3 + 12-2*5 + 12-2*6 = 10 + 8 + 6 + 2 = 26. - _Wesley Ivan Hurt_, Mar 25 2018
%H A294062 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A294062 a(n) = 2 * Sum_{i=1..n} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).
%F A294062 a(n) = 2*(n*A013928(n) - A066779(n)). - _Wesley Ivan Hurt_, Jul 08 2025
%e A294062 For n = 4, 8 can be partitioned into two parts with the smaller part squarefree in three ways: 7 + 1, 6 + 2, and 5 + 3, so a(4) = (7 - 1) + (6 - 2) + (5 - 3) = 12. - _Michael B. Porter_, Mar 27 2018
%t A294062 Table[2*Sum[(n - i) MoebiusMu[i]^2, {i, n}], {n, 80}]
%o A294062 (PARI) a(n) = 2 * sum(i=1, n, (n-i)*issquarefree(i)); \\ _Michel Marcus_, Mar 26 2018
%Y A294062 Cf. A008683 (mu), A008966, A013928, A066779.
%K A294062 nonn,easy
%O A294062 1,2
%A A294062 _Wesley Ivan Hurt_, Oct 22 2017