A273498 - OEIS

%I #20 Jun 05 2016 22:50:47

%S 2,65,72,128,468,730,793,1241,1332,1458,2000,2745,3528,4097,4160,4608,

%T 4825,5096,5840,5913,6344,8125,8192,9000,9325,9928,12168,13357,13498,

%U 14824,15626,15633,15689,16354,17640,18369,18737,19721,19773,21953,22681,27792,29449

%N Numbers that are, at the same time, the sum of: two positive squares, a positive square and a positive cube, and two positive cubes. In other words, intersection of A000404, A003325 and A055394.

%C Numbers n such that n = x^a + y^b where x,y > 0, is soluble for all 1 < a <= b < 4.

%C Perfect power terms are 128, 8192, 97344, 140625, 524288, 1500625, ...

%H Charles R Greathouse IV, <a href="/A273498/b273498.txt">Table of n, a(n) for n = 1..10000</a>

%e 793 is a term because 793 = 3^2 + 28^2 = 8^2 + 9^3 = 4^3 + 9^3.

%o (PARI) isA003325(n)=for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))

%o isA000404(n) = for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))

%o isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0

%o lista(nn) = for(n=1, nn, if(isA003325(n) && isA000404(n) && isA055394(n), print1(n, ", ")));

%o (PARI) isA000404(n)=my(f=factor(n)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); n>1 && (vecmin(f[,1]%4)==1 || (f[1, 1]==2 && f[1,2]%2))

%o isA055394(n) = for(k=1, sqrtnint(n-1,3), if(issquare(n-k^3), return(1))); 0

%o list(lim)=my(v=List(),n3,t); lim\=1; for(n=1,sqrtnint(lim-1,3), n3=n^3; for(m=1,sqrtnint(lim-n3,3), t=n3+m^3; if(isA000404(t) && isA055394(t), listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, May 31 2016

%Y Cf. A000404, A003325, A055394.

%K nonn,easy

%O 1,1

%A _Altug Alkan_, May 23 2016