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

1,1

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, ...

Links

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

Example

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

Prog

(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

Crossrefs

Cf. A000404, A003325, A055394.

Sequence in context: A287649 A229352 A229815 * A003358 A303376 A041511

Adjacent sequences: A273495 A273496 A273497 * A273499 A273500 A273501

Keyword

nonn,easy

Author

Altug Alkan, May 23 2016

Status

approved