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A125084 Cubes which have a partition as the sum of 3 squares. 5
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; text; internal format)
OFFSET
1,3
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, Nov 23 2006
A number n^3 belongs to this sequence if and only if n is sum of three squares. Proof is immediate from Catalan's identity (x^2 + y^2 + z^2)^3 = x^2*(3*z^2 - x^2 - y^2)^2 + y^2*(3*z^2 - x^2 - y^2)^2 + z^2*(z^2 - 3*x^2 - 3*y^2)^2. - Artur Jasinski, Dec 09 2006
If n = a^2 + b^2 + c^2, then n^3 = (n*a)^2 + (n*b)^2 + (n*c)^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. - Ray Chandler, Feb 03 2009
LINKS
FORMULA
a(n) = A000378(n)^3.
Equals A000578 INTERSECT A000378.
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, so 27 is a term.
125 = 5^3 = 0^2 + 2^2 + 11^2, so 125 is a term.
216 = 6^3 = 2^2 + 4^2 + 14^2, so 216 is a term.
MATHEMATICA
Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* Ray Chandler, Nov 23 2006 *)
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, ", ") ; ) ; ) ; } // R. J. Mathar, Nov 23 2006
CROSSREFS
Sequence in context: A014187 A050750 A100571 * A052048 A052064 A352049
KEYWORD
nonn
AUTHOR
Artur Jasinski, Nov 20 2006, Nov 21 2006, Nov 22 2006
EXTENSIONS
Corrected and extended by Farideh Firoozbakht, Ray Chandler and R. J. Mathar, Nov 23 2006
STATUS
approved

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Last modified February 25 02:17 EST 2024. Contains 370308 sequences. (Running on oeis4.)