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A294093 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part squarefree and smaller part prime. 1
0, 0, 0, 6, 4, 10, 8, 28, 24, 30, 36, 64, 52, 64, 28, 70, 72, 122, 88, 152, 136, 182, 164, 188, 160, 136, 192, 176, 236, 304, 240, 358, 252, 318, 328, 446, 356, 454, 376, 432, 392, 538, 364, 532, 604, 470, 508, 782, 604, 660, 680, 684, 676, 966, 736, 856 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 at prime values of x such that 2n-x is squarefree for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(8), the prime values of x that make 16-x squarefree are x=2,3,5 and so a(8) = 16-2*2 + 16-2*3 + 16-2*5 = 12 + 10 + 6 = 28. - Wesley Ivan Hurt, Mar 25 2018

LINKS

Table of n, a(n) for n=1..56.

Index entries for sequences related to partitions

FORMULA

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

EXAMPLE

For n = 9, 18 can be partitioned into a squarefree number and a smaller prime in three ways: 15 + 3, 13 + 5, and 11 + 7, so a(9) = (15 - 3) + (13 - 5) + (11 - 7) = 24. - Michael B. Porter, Mar 27 2018

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A010051, A008966, A294094.

Sequence in context: A075450 A145979 A182164 * A257926 A116946 A141270

Adjacent sequences:  A294090 A294091 A294092 * A294094 A294095 A294096

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Oct 22 2017

STATUS

approved

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Last modified October 22 07:06 EDT 2021. Contains 348160 sequences. (Running on oeis4.)