A273498 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.

2, 65, 72, 128, 468, 730, 793, 1241, 1332, 1458, 2000, 2745, 3528, 4097, 4160, 4608, 4825, 5096, 5840, 5913, 6344, 8125, 8192, 9000, 9325, 9928, 12168, 13357, 13498, 14824, 15626, 15633, 15689, 16354, 17640, 18369, 18737, 19721, 19773, 21953, 22681, 27792, 29449

Offset: 1

Author

Altug Alkan, May 23 2016

Keywords

• nonn
• easy

Comments

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

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

Examples

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000404, A003325, A055394.

Programs

PARI

isA003325(n)=for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))
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))
isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
lista(nn) = for(n=1, nn, if(isA003325(n) && isA000404(n) && isA055394(n), print1(n, ", ")));

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