cp's OEIS Frontend

A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.

%I A007692 M5299 #51 Jul 08 2025 17:05:48
%S A007692 50,65,85,125,130,145,170,185,200,205,221,250,260,265,290,305,325,338,
%T A007692 340,365,370,377,410,425,442,445,450,481,485,493,500,505,520,530,533,
%U A007692 545,565,578,580,585,610,625,629,650,680,685,689,697,725,730,740,745,754,765
%N A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.
%C A007692 A025426(a(n)) > 1. - _Reinhard Zumkeller_, Aug 16 2011
%C A007692 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 A007692 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 A007692 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 A007692 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007692 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 125.
%H A007692 Reinhard Zumkeller, <a href="/A007692/b007692.txt">Table of n, a(n) for n = 1..10000</a>
%H A007692 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 A007692 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 A007692 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A007692 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A007692 50 is a term since 1^2 + 7^2 and 5^2 + 5^2 equal 50.
%e A007692 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 A007692 625 is a term since it is the sum of squares of {0,25}, {7,24}, and {15,20}.
%t A007692 Select[Range@ 800, Length@ Select[PowersRepresentations[#, 2, 2], First@ # != 0 &] > 1 &] (* _Michael De Vlieger_, May 16 2016 *)
%o A007692 (Haskell)
%o A007692 import Data.List (findIndices)
%o A007692 a007692 n = a007692_list !! (n-1)
%o A007692 a007692_list = findIndices (> 1) a025426_list
%o A007692 -- _Reinhard Zumkeller_, Aug 16 2011
%o A007692 (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 A007692 (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 A007692 Subsequence of A001481. A subsequence is A025285 (2 ways).
%Y A007692 Cf. A004431, A118882, A000404, A018825, A025284 (one way).
%K A007692 nonn,easy
%O A007692 1,1
%A A007692 _N. J. A. Sloane_