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

1,1

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)

References

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

Links

Table of n, a(n) for n=1..37.

Example

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

Prog

(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

Crossrefs

Cf. A003828, A365688.

Sequence in context: A051980 A020271 A290338 * A254896 A165384 A365688

Adjacent sequences: A365654 A365655 A365656 * A365658 A365659 A365660

Keyword

nonn

Author

Jud McCranie, Sep 14 2023

Status

approved