

A007692


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


16



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

A025426(a(n)) > 1.  Reinhard Zumkeller, Aug 16 2011
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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
AskNRICH Archive, Numbers expressible as the sum of 2 squares in more than one way
D. J. C. Mackay and S. Mahajan, Numbers that are Sums of Squares in Several Ways
G. Xiao, Two squares
Index entries for sequences related to sums of squares


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
 Reinhard Zumkeller, Aug 16 2011
(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

Subsequence of A001481. A subsequence is A025285 (2 ways).
Cf. A004431, A118882, A000404, A018825, A025284 (one way).
Sequence in context: A215468 A109552 A206263 * A025285 A092541 A335233
Adjacent sequences: A007689 A007690 A007691 * A007693 A007694 A007695


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



