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%I #19 Dec 30 2019 10:14:00
%S 325,425,650,725,845,850,925,975,1025,1275,1300,1325,1445,1450,1525,
%T 1690,1700,1825,1850,1950,2050,2175,2225,2275,2425,2525,2535,2550,
%U 2600,2650,2725,2775,2825,2873,2890,2900,2925,2975
%N Numbers n that are the hypotenuse of exactly 7 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 7 ways.
%C Comment from _R. J. Mathar_, Feb 26 2008, edited by _Zak Seidov_ May 12 2008: (Start)
%C There are nonsquares x which can be written as a sum of 2 nonzero squares in exactly 7 different ways and which are by definition not in this sequence.
%C 203125 = (125*sqrt(13))^2 is the first example: 203125 = 625 + 202500 = 10404 + 192721 = 18225 + 184900= 22500 + 180625= 62500 + 140625= 69169 + 133956= 84100 + 119025.
%C The second and third examples are 265625 = (125*sqrt(17))^2 and 406250=(125*sqrt(26))^2. (End)
%C If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - _Ray Chandler_, Dec 30 2019
%H Chai Wah Wu, <a href="/A097101/b097101.txt">Table of n, a(n) for n = 1..10000</a>
%F Equals {n: A025426(n^2)=7}.
%e Example supplied by _R. J. Mathar_, Feb 26 2008:
%e The smallest number that can be written as a sum of two nonzero squares in 7 different ways is 105625 = 325^2:
%e 1296 + 104329 = 105625 = 325^2
%e 6400 + 99225 = 105625 = 325^2
%e 8281 + 97344 = 105625 = 325^2
%e 15625 + 90000 = 105625 = 325^2
%e 27225 + 78400 = 105625 = 325^2
%e 38025 + 67600 = 105625 = 325^2
%e 41616 + 64009 = 105625 = 325^2.
%t r[a_]:={b,c}/.{ToRules[Reduce[0<b<c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[Range[3000], Length[r[#]] == 7 &] (* _Vincenzo Librandi_, Mar 01 2016 *)
%Y Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
%K nonn
%O 1,1
%A _James R. Buddenhagen_, Sep 15 2004
%E Definition and comments corrected by _Zak Seidov_, Feb 26 2008, May 12 2008
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