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A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.
(Formerly M5299)
16

%I M5299

%S 50,65,85,125,130,145,170,185,200,205,221,250,260,265,290,305,325,338,

%T 340,365,370,377,410,425,442,445,450,481,485,493,500,505,520,530,533,

%U 545,565,578,580,585,610,625,629,650,680,685,689,697,725,730,740,745,754,765

%N Numbers that are the sum of 2 nonzero squares in 2 or more ways.

%C A025426(a(n)) > 1. - _Reinhard Zumkeller_, Aug 16 2011

%C For the question that is in the link AskNRICH Archive: It is easy to show that (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2 = (a*d + b*c)^2 + (a*c - b*d)^2. So terms of this sequence can be generated easily. 5 is the least number of the form a^2 + b^2 where a and b distinct positive integers and this is a list sequence. This is the why we observe that there are many terms which are divisible by 5. - _Altug Alkan_, May 16 2016

%C Square roots of square terms: {25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, ...}. They are also listed by A009177. - _Michael De Vlieger_, May 16 2016

%D Ming-Sun Li, Kathryn Robertson, Thomas J. Osler, Abdul Hassen, Christopher S. Simons and Marcus Wright, "On numbers equal to the sum of two squares in more than one way", Mathematics and Computer Education, 43 (2009), 102 - 108.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 125.

%H Reinhard Zumkeller, <a href="/A007692/b007692.txt">Table of n, a(n) for n = 1..10000</a>

%H AskNRICH Archive, <a href="https://web.archive.org/web/20190407133230/https://nrich.maths.org/askedNRICH/edited/76.html">Numbers expressible as the sum of 2 squares in more than one way</a>

%H D. J. C. Mackay and S. Mahajan, <a href="https://inference.org.uk/mackay/abstracts/sumsquares.html">Numbers that are Sums of Squares in Several Ways</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%e 50 is a term since 1^2 + 7^2 and 5^2 + 5^2 equal 50.

%e 25 is not a term since though 3^2 + 4^2 = 25, 25 is square, i.e., 0^2 + 5^2 = 25, leaving it with only one possible sum of 2 nonzero squares.

%e 625 is a term since it is the sum of squares of {0,25}, {7,24}, and {15,20}.

%t Select[Range@ 800, Length@ Select[PowersRepresentations[#, 2, 2], First@ # != 0 &] > 1 &] (* _Michael De Vlieger_, May 16 2016 *)

%o (Haskell)

%o import Data.List (findIndices)

%o a007692 n = a007692_list !! (n-1)

%o a007692_list = findIndices (> 1) a025426_list

%o -- _Reinhard Zumkeller_, Aug 16 2011

%o (PARI) isA007692(n)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb >= 2; \\ _Altug Alkan_, May 16 2016

%o (PARI) is(n)=my(t); if(n<9, return(0)); for(k=sqrtint(n\2-1)+1,sqrtint(n-1), if(issquare(n-k^2)&&t++>1, return(1))); 0 \\ _Charles R Greathouse IV_, Jun 08 2016

%Y Subsequence of A001481. A subsequence is A025285 (2 ways).

%Y Cf. A004431, A118882, A000404, A018825, A025284 (one way).

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_.

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Last modified January 17 23:09 EST 2021. Contains 340249 sequences. (Running on oeis4.)