A046048 -id:A046048 - cp's OEIS frontend

A186680 Total number of positive integers below 10^n requiring 17 positive biquadrates in their representation as sum of biquadrates.

0, 3, 33, 63, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65

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) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + a(n) + A186682(n) + A186684(n) = A002283(n)
a(n) = 65 for n >= 5. - Nathaniel Johnston, May 09 2011
Continued fraction expansion of (826055+sqrt(4229))/2503382. - Bruno Berselli, May 10 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046048, A099591.

PadRight[{0, 3, 33, 63}, 100, 65] (* Paolo Xausa, Jul 31 2024 *)

G.f.: x^2*(3+30*x+30*x^2+2*x^3)/(1-x). - Bruno Berselli, May 10 2011

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
Terms after a(7) from Nathaniel Johnston, May 09 2011

A186681 Total number of n-digit numbers requiring 17 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 3, 30, 30, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Feb 25 2011

A161905(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + a(n) + A186683(n) + A186685(n) = A052268(n)

a(n) = 0 for n >= 6. - Nathaniel Johnston, May 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046048, A099591, A186683, A186685.

a(n) = A186680(n) - A186680(n-1).

A099591 Numbers that are the sum of no fewer than 17 biquadrates (4th powers).

Original entry on oeis.org

47, 62, 63, 77, 78, 79, 127, 142, 143, 157, 158, 159, 207, 222, 223, 237, 238, 239, 287, 302, 303, 317, 318, 319, 367, 382, 383, 397, 398, 399, 447, 462, 463, 477, 478, 479, 527, 542, 543, 557, 558, 559, 607, 622, 623, 687, 702, 703, 752, 767, 782, 783

Offset: 1

Ralf Stephan, Oct 25 2004

There are 96 members in the sequence, the largest being 13792, see the Deshouillers et al. references.

			62 is the sum of 17 4th powers and no fewer, so 62 is a member.
63 is the sum of 18 4th powers and no fewer, so 63 is a member, although it is not a member of A046048.

T. D. Noe, Table of n, a(n) for n = 1..96 (complete sequence)
J.-M. Deshouillers, F. Hennecart and B. Landreau, Waring's Problem for sixteen biquadrates - numerical results, Journal de Théorie des Nombres de Bordeaux 12 (2000), pp. 411-422.
J.-M. Deshouillers, K. Kawada and T. D. Wooley, On Sums of Sixteen Biquadrates, Mém. Soc. Math. de France, Paris, 2005.
Eric Weisstein's World of Mathematics, Biquadratic Number
Eric Weisstein's World of Mathematics, Warings Problem

Cf. A002377, A079611, A046048.

f[n_] := f[n] = (k = 0; While[k++; PowersRepresentations[n, k, 4] == {}]; k); Select[Range[800], f[#] >= 17 &] (* Jean-François Alcover, Sep 02 2011 *)

a(25) changed from 368 to 367 by T. D. Noe, Sep 07 2006

A046047 Sum of 16 but no fewer nonzero fourth powers.

Original entry on oeis.org

31, 46, 61, 76, 111, 126, 141, 156, 191, 206, 221, 236, 271, 286, 301, 316, 351, 366, 381, 396, 431, 446, 461, 476, 496, 511, 526, 541, 556, 591, 606, 621, 671, 686, 701, 736, 751, 766, 781, 831, 846, 861, 911, 926, 941, 976, 991, 1006, 1021, 1071, 1086

Offset: 1

Eric W. Weisstein

nonn

What is the least k such that the sequence "Sum of k but no fewer nonzero fourth powers." is finite? - David A. Corneth, Jun 24 2018

13792 is the last number requiring 17 nonzero fourth powers. This sequence is infinite since numbers of the form 31*16^e always require 16 but no fewer. - Jianing Song, Jul 08 2018

David A. Corneth, Table of n, a(n) for n = 1..681 (terms <= 200000)
Eric Weisstein's World of Mathematics, Biquadratic Number
Wikipedia, Waring's problem

Cf. A000583, A002377, A046046, A046048, A099591.

Select[Range[  1100], (pr = PowersRepresentations[#, 16, 4]; test = pr != {} && Count[pr, r_ /; (Times @@ r) == 0] == 0; If[test, Print[#]]; test) &] (* Jean-François Alcover, Oct 30 2012 *)

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

A046049 Sum of 18 but no fewer nonzero fourth powers.

Original entry on oeis.org

63, 78, 143, 158, 223, 238, 303, 318, 383, 398, 463, 478, 543, 558, 623, 703, 783, 863, 943, 1008, 1023, 1103, 1183, 1248

Offset: 1

Eric W. Weisstein

See A099591 for references that show that this sequence is full.

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583, A002377, A046048, A099591.

Select[ Range[1300], (pr = PowersRepresentations[#, 18, 4]; test = pr != {} && FreeQ[pr, r_List /; (Times @@ r) == 0]; If[test, Print[#]]; test) &] (* Jean-François Alcover, Oct 30 2012 *)

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

cp's OEIS Frontend

A186680 Total number of positive integers below 10^n requiring 17 positive biquadrates in their representation as sum of biquadrates.

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A186681 Total number of n-digit numbers requiring 17 positive biquadrates in their representation as sum of biquadrates.

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A099591 Numbers that are the sum of no fewer than 17 biquadrates (4th powers).

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A046047 Sum of 16 but no fewer nonzero fourth powers.

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A046049 Sum of 18 but no fewer nonzero fourth powers.

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