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

1,2

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

Links

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

Example

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

Prog

(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

Crossrefs

Cf. A180241, A255830, A256091, A256613.

Sequence in context: A275108 A005457 A005453 * A187966 A280584 A225649

Adjacent sequences: A180239 A180240 A180241 * A180243 A180244 A180245

Keyword

nonn

Author

Carmine Suriano, Aug 19 2010

Extensions

Name clarified by David A. Corneth, Mar 20 2023

Status

approved