A046044 -id:A046044 - cp's OEIS frontend

A186671 Total number of positive integers below 10^n requiring 13 positive biquadrates in their representation as sum of biquadrates.

0, 6, 66, 706, 6945, 67173, 667369, 6667582, 66667813, 666668052, 6666668292

Offset: 1

Martin Renner, Feb 25 2011

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + a(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A046044, A186672.

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186672 Total number of n-digit numbers requiring 13 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 60, 640, 6239, 60228, 600196, 6000213, 60000231, 600000239, 6000000240

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + a(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A046044, A186671.

a(n) = A186671(n) - A186671(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A046045 Sum of 14 but no fewer nonzero fourth powers.

Original entry on oeis.org

14, 29, 44, 59, 74, 94, 109, 124, 139, 154, 174, 189, 204, 219, 224, 234, 254, 269, 284, 299, 314, 334, 349, 364, 379, 394, 414, 429, 444, 459, 464, 474, 494, 509, 524, 539, 554, 574, 589, 604, 619, 638, 654, 669, 684, 699, 704, 718, 734, 749, 764, 779, 798

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583, A002377, A046044, A047725.

More terms from Arlin Anderson (starship1(AT)gmail.com)

A047723 Sum of 12 but no fewer nonzero fourth powers.

Original entry on oeis.org

12, 27, 42, 57, 72, 92, 107, 122, 137, 152, 172, 187, 192, 202, 217, 232, 252, 267, 282, 297, 312, 332, 347, 362, 377, 392, 412, 427, 432, 442, 457, 472, 492, 497, 507, 522, 537, 552, 572, 587, 602, 617, 636, 652, 667, 672, 682, 697, 716, 732, 747, 762

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

David A. Corneth, Table of n, a(n) for n = 1..13916 (terms <= 200000)

Cf. A000583, A002377, A003346, A046044.

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(limit):
    qd = list(takewhile(lambda x: x <= limit, (k**4 for k in count(1))))
    ss = [set(sum(c) for c in mc(qd, i)) for i in range(13)]
    for i in range(11, 0, -1): ss[12] -= ss[i]
    return sorted(s for s in ss[12] if s <= limit)
print(aupto(762)) # Michael S. Branicky, Dec 27 2021

Typo in data corrected by D. S. McNeil, Aug 17 2010

A047724 Sum of 13 nonzero fourth powers.

Original entry on oeis.org

13, 28, 43, 58, 73, 88, 93, 103, 108, 118, 123, 133, 138, 148, 153, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 228, 233, 243, 248, 253, 258, 263, 268, 273, 278, 283, 293, 298, 308, 313, 323, 328, 333, 338, 343, 348, 358, 363, 373, 378, 388

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

Cf. A000583, A003346, A046044.

Union[Total/@Tuples[Range[3]^4,13]] (* Harvey P. Dale, Nov 04 2015 *)

A085304 Least number of 4th powers required to represent n!.

Original entry on oeis.org

1, 1, 2, 6, 9, 10, 15, 15, 9, 10, 15, 6, 12, 12

Offset: 0

Labos Elemer, Jun 30 2003

			n=6: 6!=720=625+81+14,length-of-solution=16>=a(6)
but 6!=720=2.256+13.16 seems shortest solution a(6)=15
after, see also A046046
n=7: 7!=5040=3.1296+4.256+8.16 so a(7)<=15 (uncertain);
n=8: a(8)<=9 because 8!=4.10000+1.256+4.16.

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000142, A084355, A060387, A003336-A003346, A046046, A046044-A046050.

"Shortest" solutions to n!=Sum[x(j)^4], j=1, .., m[n] with minimal value of m[n]: a(n)=Min{m[n]}. Per analogiam A084355.

a(7)-a(11) from John W. Layman, Aug 13 2004
a(12) from Sean A. Irvine, Feb 11 2010
a(13) from Sean A. Irvine, Feb 15 2010

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A186671 Total number of positive integers below 10^n requiring 13 positive biquadrates in their representation as sum of biquadrates.

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A186672 Total number of n-digit numbers requiring 13 positive biquadrates in their representation as sum of biquadrates.

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A046045 Sum of 14 but no fewer nonzero fourth powers.

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A047723 Sum of 12 but no fewer nonzero fourth powers.

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A047724 Sum of 13 nonzero fourth powers.

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A085304 Least number of 4th powers required to represent n!.

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