Search: id:A080761
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%I A080761
%S A080761 1,3,8,9,12,15,17,18,19,22,24,28,30,35,36,37,38,40,41,44,48,54,55,56,57,
%T A080761 63,64,65,68,71,73,79,80,89,92,94,97,98,99,100,101,105,106,107,108,112,
%U A080761 113,117,119,120,121,128,129,131,132,136,138,141,142,143,145,148,151
%N A080761 Conjectured list of positive numbers of the form y^2 - x^3, x and y >
               = 1.
%C A080761 Obtained by considering only values of x <= 10^6.
%C A080761 Comments from Artur Jasinski (grafix(AT)csl.pl), Oct 03 2007 (Start): 
               Some numbers have multiple partitions:
%C A080761 8 = 4^2 - 8^3 = 312^2 - 46^39
%C A080761 9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^2
%C A080761 17 = 5^2 - 2^3 = 9^2 - 4^3 = 23^2 - 8^3 = 282^2 - 43^3 = 365^2 - 52^2 
               = 378661^2-5234^3 (First squared factor: A029727, Second cubed factor: 
               A029728)
%C A080761 24 = 32^2 - 10^2 = 736844^2 - 8158^3
%C A080761 36 = 10^2 - 4^3 = 42^2 - 12^3
%C A080761 57 = 11^2 - 4^3 = 20^2 - 7^3
%C A080761 65 = 53^2 - 14^3 = 14113^2 - 584^3
%C A080761 73 = 17^2 - 6^3 = 611^2 - 72^3 = 6717^2 - 365^3
%C A080761 89 = 33^2 - 10^3 = 408^2 - 55^3
%C A080761 100 = 90^2 - 20^3 = 118^2 - 24^3 = 137190^2 - 2660^3
%C A080761 108 = 18^2 - 6^3 = 7002^2 - 366^3
%C A080761 113 = 25^2 - 8^3 = 38^2 - 11^3 = 133^2 - 26^3 = 8669^2 - 422^3 (First 
               squared factor: A134042 Second cubed factor: A134043
%C A080761 141 = 22^2 - 7^3 = 169852^2 - 3067^3
%H A080761 Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/n37ltk2.msnw">Proof 
               that n^3+7 <> k^2 for all integers n,k</a>.
%e A080761 8 is in the sequence since 3^2 = 1^3 + 8.
%o A080761 (PARI) diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),
               print1(p" "); break) ) ) }
%Y A080761 Complement of A080762.
%Y A080761 Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
%Y A080761 Cf. A029727, A029728, A134042, A134043.
%Y A080761 Sequence in context: A028960 A139491 A084387 this_sequence A087286 A165289 
               A066494
%Y A080761 Adjacent sequences: A080758 A080759 A080760 this_sequence A080762 A080763 
               A080764
%K A080761 nonn
%O A080761 1,2
%A A080761 Cino Hilliard (hillcino368(AT)gmail.com), Mar 10 2003
%E A080761 "Positive" added to definition by N. J. A. Sloane (njas(AT)research.att.com), 
               Oct 06 2007

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