A365657 Integers k such that k^2 can be written as the sum of three positive fourth powers.

481, 1924, 4329, 7696, 12025, 17316, 23569, 24961, 28721, 30784, 38961, 48100, 58201, 65441, 69121, 69264, 81289, 94276, 99844, 108225, 113241, 114884, 123136, 139009, 155844, 173641, 192400, 212121, 224649, 232804, 254449, 258489, 261764, 276484, 277056, 300625, 325156

Offset: 1

Author

Jud McCranie, Sep 14 2023

Keywords

• nonn

Comments

Primitive solutions are in A365688.

From Jon E. Schoenfield, Sep 15 2023: (Start)

If k is a term, then so is m^2 * k for every m > 1.

Every even term is four times a smaller term.

Every odd term is the square root of the sum of one odd fourth power and two even fourth powers.

(End)

Examples

			1924^2 = 24^4 + 30^4 + 40^4.
		

References

• Jean-Marie De Koninck, "Those Fascinating Numbers", AMS, 2008, entry 481.

Crossrefs

Cf. A003828, A365688.

Programs

Python

from itertools import count, islice
from sympy import integer_nthroot
def A365657_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        m, flag = k**2, False
        for x in count(1):
            if (x4:=x**4)+2>m or flag:
                break
            for y in range(min(x,integer_nthroot(m-x4-1,4)[0]),0,-1):
                if (z4:=m-x4-(y4:=y**4))>y4 or flag:
                    break
                if integer_nthroot(z4,4)[1]:
                    yield k
                    flag = True
                    break
A365657_list = list(islice(A365657_gen(),6)) # Chai Wah Wu, Sep 19 2023