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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
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 9 16:24 EDT 2020. Contains 335544 sequences. (Running on oeis4.)