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%I #22 May 09 2021 11:20:08
%S 160,378,476,495,1366,1464,1483,1682,1701,1799,2232,2330,2349,2548,
%T 2567,2665,3536,3555,3653,3871,4948,5046,5065,5264,5283,5381,6252,
%U 6271,6369,6587,6894,6992,7011,7118,7137,7210,7229,7235,7327,7453,8198,8217,8315
%N Numbers which are the sum of three cubes of distinct primes.
%C This is a subsequence of A024975. The odd terms of this sequence are A138853, the even terms are 8+{ even terms of A120398 }. Thus primes in this sequence, A137365, are the same as primes in A138853.
%H Chai Wah Wu, <a href="/A138854/b138854.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..621 from R. J. Mathar)
%H <a href="/index/Su#ssq">Index to sequences related to sums of cubes</a>.
%F A138854 = { p(i)^3+p(j)^3+p(k)^3 ; i>j>k>0 } = A138853 union { p(i)^3+p(j)^3+8 ; i>j>1}
%p isA030078 := proc(n)
%p local f ;
%p if n < 8 then
%p false;
%p else
%p f := ifactors(n)[2] ;
%p if nops(f) = 1 and op(2,op(1,f)) = 3 then
%p true;
%p else
%p false;
%p end if;
%p end if;
%p end proc:
%p isA138854 := proc(n)
%p local i,j,p,q,r,rcub ;
%p for i from 1 do
%p p := ithprime(i) ;
%p if p^3+(p+1)^3+(p+2)^3 > n then
%p return false;
%p end if;
%p for j from i+1 do
%p q := ithprime(j) ;
%p rcub := n-q^3-p^3 ;
%p if rcub <= q^3 then
%p break;
%p fi ;
%p if isA030078(rcub) then
%p return true;
%p end if;
%p end do:
%p end do:
%p end proc:
%p for n from 5 do
%p if isA138854(n) then
%p print(n);
%p end if;
%p end do: # _R. J. Mathar_, Jun 09 2014
%t f[upto_]:=Module[{maxp=PrimePi[Floor[Power[upto, (3)^-1]]]}, Select[Union[Total/@(Subsets[Prime[Range[maxp]],{3}]^3)],#<=upto&]]; f[9000] (* _Harvey P. Dale_, Mar 21 2011 *)
%o (PARI) isA138854(n)={ if( n%2, isA138853(n), isA120398(n-8)) }
%o for( n=1,10^4, isA138854(n) & print1(n", "))
%Y Cf. A024975 (a^3+b^3+c^3, a>b>c>0), A138853 (odd terms of this), A120398, A137365 (primes in A138853 / A138854).
%K nonn
%O 1,1
%A _M. F. Hasler_, Apr 13 2008