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A294094 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part prime and smaller part squarefree. 1

%I #19 Apr 16 2018 19:00:24

%S 0,2,4,8,4,12,20,16,28,38,28,48,32,24,56,64,68,60,68,58,112,144,104,

%T 168,124,110,180,124,152,202,192,224,204,190,188,288,344,288,300,300,

%U 304,398,344,290,464,326,384,360,304,418,540,556,444,616,608,626,764

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

%C 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 such that 2n-x is prime 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 that make 12-x prime are x=1,5 and so a(6) = 12-2*1 + 12-2*5 = 10 + 2 = 12. - _Wesley Ivan Hurt_, Mar 25 2018

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

%F a(n) = 2 * Sum_{i=1..n} (n - i) * A010051(2n - i) * A008966(i).

%e For n = 7, 14 can be partitioned into a prime and a smaller squarefree number in two ways: 13 + 1 and 11 + 3, so a(7) = (13 - 1) + (11 - 3) = 20. - _Michael B. Porter_, Mar 27 2018

%t Table[2*Sum[(n - i) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]) MoebiusMu[i]^2, {i, n}], {n, 80}]

%o (PARI) a(n) = 2 * sum(i=1, n, (n-i)*isprime(2*n-i)*issquarefree(i)); \\ _Michel Marcus_, Mar 26 2018

%Y Cf. A010051, A008966, A294093.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 22 2017

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)