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A009003
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Hypotenuse numbers (squares are sums of 2 nonzero squares).
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53
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5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 140
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OFFSET
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1,1
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COMMENTS
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Multiples of Pythagorean primes A002144 or of primitive Pythagorean triangles' hypotenuses A008846. - Lekraj Beedassy, Nov 12 2003
It appears that this is exactly the sequence of positive integers with at least one prime divisor of the form 4k + 1. (This has been verified for all terms less than or equal to 500.) Compare A072592. - John W. Layman, Mar 12 2008
The conjecture by Layman is correct. It is well known that the hypotenuses of primitive Pythagorean triples are precisely those numbers with all prime divisors of the form 4k + 1. - Franklin T. Adams-Watters, Apr 26 2009
Circumradius R of the triangles such that the area, the sides and R are integers. - Michel Lagneau, Mar 03 2012
The 2 squares summing to a(n)^2 cannot be equal because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 10 2013
Closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 4k + 1 with multiplicity one, which are also those for which there exists a unique integer triangle = A084645. - Jean-Christophe Hervé, Nov 11 2013
a(n) are numbers whose square is the mean of two distinct nonzero squares. This creates 1-to-1 mapping between a Pythagorean triple and a "Mean" triple. If the Pythagorean triple is written, abnormally, as {j, k, h} where j^2 +(j+k)^2 = h^2, and h = a(n), then the corresponding "Mean" triple with the same h is {k, 2j, h} where (k^2 + (k+2j)^2)/2 = h^2. For example for h = 5, the Pythagorean triple is {3, 1, 5} and the Mean triple is {1, 6, 5}. - Richard R. Forberg, Mar 01 2015
Integral side lengths of rhombuses with integral diagonals p and q (therefore also with integral areas A because A = pq/2 is some multiple of 24). No such rhombuses are squares. - Rick L. Shepherd, Apr 09 2017
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
R. Chapman, Pythagorean triples and sums of squares
S. R. Finch, Landau-Ramanujan Constant
Ron Knott, Pythagorean Triples and Online Calculators
J. Pahikkala, On contraharmonic mean and Pythagorean triples, Elemente der Mathematik, 65:2 (2010), 62-67.
Index entries for sequences related to sums of squares
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FORMULA
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A005089(a(n)) > 0. - Reinhard Zumkeller, Jan 07 2013
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MAPLE
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isA009003 := proc(n)
local p;
for p in numtheory[factorset](n) do
if modp(p, 4) = 1 then
return true;
end if;
end do:
false;
end proc:
for n from 1 to 200 do
if isA009003(n) then
printf("%d, ", n) ;
end if;
end do: # R. J. Mathar, Nov 17 2014
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MATHEMATICA
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f[n_] := Module[{k = 1}, While[(n - k^2)^(1/2) != IntegerPart[(n - k^2)^(1/2)], k++; If[2 * k^2 >= n, k = 0; Break[]]]; k]; A009003 = {}; Do[If[f[n^2] > 0, AppendTo[A009003, n]], {n, 3, 100}]; A009003 (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)
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PROG
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(PARI) is_A009003(n)=setsearch(Set(factor(n)[, 1]%4), 1) \\ M. F. Hasler, May 27 2012
(Haskell)
import Data.List (findIndices)
a009003 n = a009003_list !! (n-1)
a009003_list = map (+ 1) $ findIndices (> 0) a005089_list
-- Reinhard Zumkeller, Jan 07 2013
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CROSSREFS
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Cf. A009000, A024507, A004431, A072592, A004613, A187811.
Complement of A004144. Primes in this sequence give A002144. Same as A146984 (integer contraharmonic means) as sets - see Pahikkala 2010, Theorem 5.
Cf. A083025, A084645 (primitive elements), A084646, A084647, A084648, A084649, A006339.
Sequence in context: A198389 A057100 A304436 * A071821 A201012 A209922
Adjacent sequences: A009000 A009001 A009002 * A009004 A009005 A009006
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Definition edited by Jean-Christophe Hervé, Nov 10 2013
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STATUS
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approved
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