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.

0, 2, 4, 8, 4, 12, 20, 16, 28, 38, 28, 48, 32, 24, 56, 64, 68, 60, 68, 58, 112, 144, 104, 168, 124, 110, 180, 124, 152, 202, 192, 224, 204, 190, 188, 288, 344, 288, 300, 300, 304, 398, 344, 290, 464, 326, 384, 360, 304, 418, 540, 556, 444, 616, 608, 626, 764

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

Formula

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

Example

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

Mathematica

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

Prog

(PARI) a(n) = 2 * sum(i=1, n, (n-i)*isprime(2*n-i)*issquarefree(i)); \\ Michel Marcus, Mar 26 2018

Crossrefs

Cf. A010051, A008966, A294093.

Sequence in context: A151569 A016635 A133992 * A290288 A126215 A165617

Adjacent sequences: A294091 A294092 A294093 * A294095 A294096 A294097

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 22 2017

Status

approved