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A125084
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Cubes which have a partition as the sum of 3 squares.
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4
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0, 1, 8, 27, 64, 125, 216, 512, 729, 1000, 1331, 1728, 2197, 2744, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 64000, 68921, 74088, 79507, 85184, 91125
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| If n is of the form 4^i*(8j+7) (where i>=0, j>=0) then n^3 is not in the sequence because n^3 is of the mentioned form so n^3 is in A004215 hence according to the definition n^3 is not in this sequence (see formula for A004215). Hence 7^3, 15^3, 23^3, 28^3, 31^3, 39^3, ... are not in the sequence. Is there a number n such that n^3 is not in the sequence but n is not of the form 4^i*(8j+7)? - Farideh Firoozbakht (mymontain(AT)yahoo.com), Nov 23 2006
A number n^3 belongs to this sequence iff and only n is sum of three squares. Proof is immediate from Catalan's identity (x^2+y^2+z^2)^3=x^2(3z^2-x^2-y^2)^2+y^2(3z^2-x^2-y^2)^2+z^2(z^2-3x^2-3y^2)^2. - Artur Jasinski (grafix(AT)csl.pl), Dec 09 2006
If n=a^2+b^2+c^2, then n^3=(na)^2+(nb)^2+(nc)^2. Conversely, suppose there were an n such that n^3 is in A000378 but n is not. Then n must be of form 4^k*(8i+7). But n^3 would also be of the form 4^k*(8i+7) and thus n^3 would not be in A000378, contradicting the original assumption. This argument is easily extended to all odd powers, i.e. n^(2k+1) is in A000378 iff n is in A000378. [From Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 03 2009]
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FORMULA
| a(n) = A000378(n)^3.
Equals A000578 INTERSECT A000378.
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EXAMPLE
| 125 is in the sequence because 125=5^3=0^2+2^2+11^2=0^2+5^2+10^2= 3^2+4^2+10^2=5^2+6^2+8^2.
27=3^3=1^2+1^2+5^2. 125=5^3=2^2+0^2+11^2. 216=6^3=4^2+2^2+14^2.
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MATHEMATICA
| Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* Chandler *)
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PROG
| (PARI) isA125084(n)={ local(cnt, a, b) ; cnt=0 ; a=0; while(a^2<=n, b=0 ; while(b<=a && a^2+b^2<=n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ; } { for(n=1, 300, if(isA125084(n^3), print1(n^3, ", ") ; ) ; ) ; } (Mathar)
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CROSSREFS
| Cf. A004215.
Sequence in context: A014187 A050750 A100571 * A052048 A052064 A125496
Adjacent sequences: A125081 A125082 A125083 * A125085 A125086 A125087
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KEYWORD
| nonn
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AUTHOR
| Artur Jasinski (grafix(AT)csl.pl), Nov 20 2006, Nov 21 2006, Nov 22 2006
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EXTENSIONS
| Corrected and extended by Farideh Firoozbakht (mymontain(AT)yahoo.com), Ray Chandler (rayjchandler(AT)sbcglobal.net) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2006
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