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Search: id:A009003
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| A009003 |
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Hypotenuse numbers (squares are sums of 2 distinct nonzero squares). |
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25
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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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Multiples of Pythagorean primes A002144 or of primitive Pythagorean triangles' hypotenuses A008846. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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<=500.) Compare A072592. - John W. Layman (layman(AT)math.vt.edu), 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. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 26 2009]
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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
Index entries for sequences related to sums of squares
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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]; lst={}; Do[If[f[n^2]>0, AppendTo[lst, n]], {n, 3, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 15 2009]
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CROSSREFS
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Cf. A009000, A009003, A024507, A004431. Complement of A004144.
Primitive elements give A002144.
Cf. A072592.
Cf. A004613 [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 26 2009]
Adjacent sequences: A009000 A009001 A009002 this_sequence A009004 A009005 A009006
Sequence in context: A049197 A009000 A057100 this_sequence A071821 A084645 A092604
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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