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A334239
Number of r X s rectangles with composite side lengths such that r + s = 2n, r <= s and r | s.
0
0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 4, 0, 2, 3, 3, 0, 5, 0, 4, 3, 2, 0, 6, 2, 2, 4, 4, 0, 7, 0, 4, 3, 2, 3, 8, 0, 2, 3, 6, 0, 7, 0, 4, 7, 2, 0, 8, 2, 5, 3, 4, 0, 8, 3, 6, 3, 2, 0, 11, 0, 2, 7, 5, 3, 7, 0, 4, 3, 7, 0, 11, 0, 2, 7, 4, 3, 7, 0, 8, 6, 2, 0, 11, 3, 2, 3, 6, 0
OFFSET
1,6
FORMULA
a(n) = Sum_{i=4..n} (1 - ceiling((2*n-i)/i) + floor((2*n-i)/i)) * (1 - c(i)) * (1 - c(2*n-i)), where c is the prime characteristic (A010051).
EXAMPLE
a(6) = 2; 2*6 = 12 has two rectangles with composite side lengths, 4 X 8 and 6 X 6. Furthermore, 4 | 8 and 6 | 6.
a(12) = 24; 2*12 = 24 has four rectangles with composite side lengths where the smallest divides the largest. They are 4 X 20, 6 X 18, 8 X 16 and 12 X 12.
MATHEMATICA
Table[Sum[(1 - Ceiling[(2 n - i)/i] + Floor[(2 n - i)/i]) (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[2 n - i] + PrimePi[2 n - i - 1]), {i, 4, n}], {n, 100}]
CROSSREFS
Cf. A010051.
Sequence in context: A127504 A321665 A329321 * A335062 A047917 A144569
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 19 2020
STATUS
approved