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A208323 "Natural" rectangular pyramidal numbers. 1
1, 2, 3, 5, 5, 8, 7, 11, 14, 14, 11, 20, 13, 20, 26, 30, 17, 32, 19, 40, 38, 32, 23, 50, 55, 38, 50, 60, 29, 70, 31, 70, 62, 50, 85, 91, 37, 56, 74, 100, 41, 112, 43, 100, 115, 68, 47, 133, 140, 130, 98, 120, 53, 154, 145, 168, 110, 86, 59, 175, 61, 92, 196 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence shows the maximum number of spheres in a pyramid with a rectangular base, where the base consists of n spheres. The area of the base n is the product of the lengths of its edges a and b, where 0 <= b <= a. In order to find the maximum number of spheres in the pyramid a(n), for a certain n we have to find factors a and b as close to each other, i.e. as close to sqrt(n), as possible. Therefore, b = A033676(n). The number b also represents the number of floors in the pyramid (i.e. its height in spheres).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = sum(i=0..b-1, (a-i)(b-i)), n=ab, 0 <= b <= a, b = A033676(n).

EXAMPLE

For n = a*b = 12, a and b must be as close to sqrt(12) as possible. Therefore, a=4, b=3 and a(n) = sum(i=0..2, (4-i)(3-i)) = 20.

For any prime number n, a(n) = n.

MATHEMATICA

Table[If[IntegerQ[Sqrt[n]], w = h = Sqrt[n], d = Divisors[n]; len = Length[d]/2; {w, h} = d[[{len, len+1}]]]; Sum[(w - i) (h - i), {i, 0, w - 1}], {n, 63}] (* T. D. Noe, Feb 28 2012 *)

CROSSREFS

Cf. A000330, A033676.

Sequence in context: A166250 A174088 A304493 * A067284 A353272 A123339

Adjacent sequences:  A208320 A208321 A208322 * A208324 A208325 A208326

KEYWORD

easy,nonn

AUTHOR

Ivan N. Ianakiev, Feb 25 2012

STATUS

approved

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Last modified May 25 00:03 EDT 2022. Contains 354047 sequences. (Running on oeis4.)