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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
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 9 00:09 EDT 2021. Contains 343685 sequences. (Running on oeis4.)