login
A180242
Numbers whose square cannot be expressed as the sum of a positive square, a positive cube and a positive fourth power.
9
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
KEYWORD
nonn
AUTHOR
Carmine Suriano, Aug 19 2010
EXTENSIONS
Name clarified by David A. Corneth, Mar 20 2023
STATUS
approved