

A125084


Cubes which have a partition as the sum of 3 squares.


4



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
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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 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^2x^2y^2)^2+y^2(3z^2x^2y^2)^2+z^2(z^23x^23y^2)^2.  Artur Jasinski, 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, Feb 03 2009]


LINKS

Table of n, a(n) for n=1..40.


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. 125=5^3=2^2+0^2+11^2. 216=6^3=4^2+2^2+14^2.


MATHEMATICA

Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* Chandler *)


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(na^2b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ; } { for(n=1, 300, if(isA125084(n^3), print1(n^3, ", ") ; ) ; ) ; } (Mathar)


CROSSREFS

Cf. A004215.
Sequence in context: A014187 A050750 A100571 * A052048 A052064 A125496
Adjacent sequences: A125081 A125082 A125083 * A125085 A125086 A125087


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



