A003346 - OEIS

%I #30 Dec 27 2021 08:25:57

%S 12,27,42,57,72,87,92,102,107,117,122,132,137,147,152,162,167,172,177,

%T 182,187,192,197,202,212,217,227,232,242,247,252,257,262,267,277,282,

%U 292,297,307,312,322,327,332,342,347,357,362,372,377,387,392,402,407,412,417

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

%C 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

%H David A. Corneth, <a href="/A003346/b003346.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number.</a>

%e From _David A. Corneth_, Aug 03 2020: (Start)

%e 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.

%e 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.

%e 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)

%o (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

%o (Python)

%o from itertools import count, takewhile, combinations_with_replacement as mc

%o def aupto(limit):

%o     qd = takewhile(lambda x: x <= limit, (k**4 for k in count(1)))

%o     ss = set(sum(c) for c in mc(qd, 12))

%o     return sorted(s for s in ss if s <= limit)

%o print(aupto(417)) # _Michael S. Branicky_, Dec 27 2021

%Y Cf. A000583 (4th powers).

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

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

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

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

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

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

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

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_