

A007692


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


20



50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 325, 338, 340, 365, 370, 377, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 680, 685, 689, 697, 725, 730, 740, 745, 754, 765
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OFFSET

1,1


COMMENTS

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
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


REFERENCES

MingSun 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 125.


LINKS



EXAMPLE

50 is a term since 1^2 + 7^2 and 5^2 + 5^2 equal 50.
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.
625 is a term since it is the sum of squares of {0,25}, {7,24}, and {15,20}.


MATHEMATICA

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


PROG

(Haskell)
import Data.List (findIndices)
a007692 n = a007692_list !! (n1)
a007692_list = findIndices (> 1) a025426_list
(PARI) isA007692(n)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((nx^2 >= x^2) && issquare(nx^2), nb++); ); nb >= 2; \\ Altug Alkan, May 16 2016
(PARI) is(n)=my(t); if(n<9, return(0)); for(k=sqrtint(n\21)+1, sqrtint(n1), if(issquare(nk^2)&&t++>1, return(1))); 0 \\ Charles R Greathouse IV, Jun 08 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



