This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A125084 #29 May 03 2024 20:18:35
%S A125084 0,1,8,27,64,125,216,512,729,1000,1331,1728,2197,2744,4096,4913,5832,
%T A125084 6859,8000,9261,10648,13824,15625,17576,19683,24389,27000,32768,35937,
%U A125084 39304,42875,46656,50653,54872,64000,68921,74088,79507,85184,91125
%N A125084 Cubes which have a partition as the sum of 3 squares.
%C A125084 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 A125084 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 A125084 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 A125084 Amiram Eldar, <a href="/A125084/b125084.txt">Table of n, a(n) for n = 1..2000</a>
%F A125084 a(n) = A000378(n)^3.
%F A125084 Equals A000578 INTERSECT A000378.
%e A125084 125 is in the sequence because
%e A125084 125 = 5^3 = 0^2 + 2^2 + 11^2
%e A125084 = 0^2 + 5^2 + 10^2
%e A125084 = 3^2 + 4^2 + 10^2
%e A125084 = 5^2 + 6^2 + 8^2.
%e A125084 27 = 3^3 = 1^2 + 1^2 + 5^2, so 27 is a term.
%e A125084 125 = 5^3 = 0^2 + 2^2 + 11^2, so 125 is a term.
%e A125084 216 = 6^3 = 2^2 + 4^2 + 14^2, so 216 is a term.
%t A125084 Select[Range[0, 50]^3, SquaresR[3, # ] > 0 &] (* _Ray Chandler_, Nov 23 2006 *)
%o A125084 (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 A125084 Cf. A000378, A000578, A004215.
%K A125084 nonn
%O A125084 1,3
%A A125084 _Artur Jasinski_, Nov 20 2006, Nov 21 2006, Nov 22 2006
%E A125084 Corrected and extended by _Farideh Firoozbakht_, _Ray Chandler_ and _R. J. Mathar_, Nov 23 2006