A003346 Numbers that are the sum of 12 positive 4th powers.

12, 27, 42, 57, 72, 87, 92, 102, 107, 117, 122, 132, 137, 147, 152, 162, 167, 172, 177, 182, 187, 192, 197, 202, 212, 217, 227, 232, 242, 247, 252, 257, 262, 267, 277, 282, 292, 297, 307, 312, 322, 327, 332, 342, 347, 357, 362, 372, 377, 387, 392, 402, 407, 412, 417

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(88) = 636 = 5^4 + 11 and a(91) = 651 = 5^4 + 2^4 + 10 are the first two terms not congruent to 2 or 7 (mod 10). - M. F. Hasler, Aug 03 2020

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
3740 is in the sequence as 3740 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 3^4 + 5^4 + 5^4 + 7^4.
4690 is in the sequence as 4690 = 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 5^4 + 5^4 + 6^4 + 6^4.
7193 is in the sequence as 7193 = 2^4 + 4^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 5^4 + 6^4. (End)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Biquadratic Number.

Crossrefs

Cf. A000583 (4th powers).

Other numbers that are the sum of k positive m-th powers:

A000404 (k=2, m=2), A000408 (3, 2), A000414 (4, 2), A047700 (k=5, m=2),

A003325 (k=2, m=3), A003072 (k=3, m=3), A003327 .. A003335 (k=4..12, m=3),

A003336 .. A003346 (k=2..12, m=4), A003347 .. A003357 (k=2..12, m=5),

A003358 .. A003368 (k=2..12, m=6), A003369 .. A003379 (k=2..12, m=7),

A003380 .. A003390 (k=2..12, m=8), A003391 .. A004801 (k=2..12, m=9),

A004802 .. A004812 (k=2..12, m=10), A004813 .. A004823 (k=2..12, m=11).

Programs

PARI

(A003346_upto(N, k=12, m=4)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(500) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

Python

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(limit):
    qd = takewhile(lambda x: x <= limit, (k**4 for k in count(1)))
    ss = set(sum(c) for c in mc(qd, 12))
    return sorted(s for s in ss if s <= limit)
print(aupto(417)) # Michael S. Branicky, Dec 27 2021