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A059267
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Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 mod (4) or have a divisor d of the form d = m^2 - 1.
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4
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3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| A099475(a(n)) > 0: complement of A099477; A008586, A008585 and A037074 are subsequences - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 18 2004
Also: values for which A143714 is nonzero, i.e. maximal side lengths of cuboids having integer shortest path going on the surface from one vertex to the opposite one (cf. example and A143714-A143715 for more). This path is the hypotenuse of a Pythagorean triple with other legs being the maximal side length and the sum of the two other side lengths of the cuboid. Thus the sequence lists integers c such that there are positive integers a <= b <= c such that c^2 + (a+b)^2 is a square (cf. PARI code). [From M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 29 2008]
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LINKS
| M. F. Hasler, Table of n, a(n) for n=1,...,3131.
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EXAMPLE
| a(18) = 35 because 5 and 7 divides 35 and 7 - 5 = 2
Contribution from M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 29 2008: (Start)
The 3 x 3 x 1 and 3 x 2 x 2 cuboids are the smallest (in terms of the longest side) for which the shortest path going on the surface from one vertex to the opposite one has integer length (namely sqrt(3^2+(3+1)^2) = sqrt(3^2+(2+2)^2) = 5), therefore the sequence lists 3 as first member.
The number 4 corresponds to the 4 x 2 x 1 cuboid and path length sqrt(4^2+(2+1)^2).
The number 5 is not part of this sequence since there is no cuboid with longest side length 5 and having integer shortest path from one vertex to the opposite one. (End)
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MAPLE
| with(numtheory): for n from 1 to 1000 do flag := 1: if n mod 4 = 0 then printf(`%d, `, n):flag := 0 fi: for m from 2 to ceil(sqrt(n)) do if n mod (m^2-1) = 0 and flag=1 then printf(`%d, `, n); break fi: od: od:
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PROG
| (PARI) isA059267(n)={ n%4==0 | fordiv( n, d, issquare(d+1) & return(1))} \\ [From M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 29 2008]
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CROSSREFS
| Cf. A143714-A143715 (number of cuboids having the cited property). [From M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Aug 29 2008]
Sequence in context: A195019 A192519 A036446 * A049433 A135251 A088070
Adjacent sequences: A059264 A059265 A059266 * A059268 A059269 A059270
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KEYWORD
| nonn
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AUTHOR
| Avi Peretz (njk(AT)netvision.net.il), Jan 23 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 24 2001
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