A125084 - OEIS

%I #29 May 03 2024 20:18:35

%S 0,1,8,27,64,125,216,512,729,1000,1331,1728,2197,2744,4096,4913,5832,

%T 6859,8000,9261,10648,13824,15625,17576,19683,24389,27000,32768,35937,

%U 39304,42875,46656,50653,54872,64000,68921,74088,79507,85184,91125

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

%C 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

%C 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

%C 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

%H Amiram Eldar, <a href="/A125084/b125084.txt">Table of n, a(n) for n = 1..2000</a>

%F a(n) = A000378(n)^3.

%F Equals A000578 INTERSECT A000378.

%e 125 is in the sequence because

%e   125 = 5^3 = 0^2 + 2^2 + 11^2

%e             = 0^2 + 5^2 + 10^2

%e             = 3^2 + 4^2 + 10^2

%e             = 5^2 + 6^2 +  8^2.

%e    27 = 3^3 = 1^2 + 1^2 +  5^2, so  27 is a term.

%e   125 = 5^3 = 0^2 + 2^2 + 11^2, so 125 is a term.

%e   216 = 6^3 = 2^2 + 4^2 + 14^2, so 216 is a term.

%t Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* _Ray Chandler_, Nov 23 2006 *)

%o (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

%Y Cf. A000378, A000578, A004215.

%K nonn

%O 1,3

%A _Artur Jasinski_, Nov 20 2006, Nov 21 2006, Nov 22 2006

%E Corrected and extended by _Farideh Firoozbakht_, _Ray Chandler_ and _R. J. Mathar_, Nov 23 2006