A180242 Numbers whose square cannot be expressed as the sum of a positive square, a positive cube and a positive fourth power.

1, 2, 3, 4, 6, 10, 11, 13, 14, 15, 16, 20, 26, 27, 29, 30, 34, 36, 40, 43, 48, 49, 50, 56, 58, 62, 64, 76, 86, 92, 94, 102, 104, 106, 122, 126, 130, 146, 148, 176, 178, 202, 211, 218, 227, 232, 238, 246, 248, 262, 272, 281, 286, 310, 311, 326, 335, 344, 346, 349, 370

Offset: 1

Author

Carmine Suriano, Aug 19 2010

Keywords

• nonn

Comments

Complement to A180241 with respect to the set of positive integers.

If k^2 = m^2 + t^3 + u^4 where k, m, t and u are positive then k^2 - m^2 = (k - m)*(k + m) = t^3 + u^4 which might ease the search for terms by looking at divisors of t^3 + u^4. - David A. Corneth, Apr 03 2023

Examples

			a(5) = 6 since 6^2 = 36 cannot be expressed as the sum of a square, a cube and a fourth power.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..3170 (terms <= 10^7)

Crossrefs

Cf. A180241, A255830, A256091, A256613.

Programs

PARI

for(D=1,99,for(C=1,sqrtn(D^2-1,4),for(B=1,sqrtn(D^2-C^4-1,3),issquare(D^2-C^4-B^3)&&next(3)));print1(D",")) \\ M. F. Hasler, Apr 06 2015

PARI

is(n)=my(n2=n^2); for(C=1, sqrtn(n2-1, 4), my(t=n2-C^4); for(B=1, sqrtn(t-1, 3), if(issquare(t-B^3), return(0)))); 1 \\ Charles R Greathouse IV, Apr 06 2015

Extensions

Name clarified by David A. Corneth, Mar 20 2023