A002377 -id:A002377 - cp's OEIS frontend

A185673 Least number k having n representations as the sum of the minimal number of biquadrates A002377(k).

1, 259, 518, 777, 3402, 3645, 3726, 7045, 7243, 12683, 16441, 13723, 13792, 21631, 20202, 23002, 24135, 27162, 28870, 28215, 33230, 39629, 36510, 41561, 43241, 29563, 47401, 41310, 47150, 47790, 56749, 43962, 48750, 62681, 65069, 50442

Offset: 1

Martin Renner, Feb 09 2011

nonn

This sequence is not monotonically increasing: a(21)=33230 > a(26)=29563.

			a(1) = 1 since 1 = 1^4 (1 way with minimal representation)
a(2) = 259 since 259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4 (2 ways with minimal representation)
a(3) = 518 since 518 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4 + 4^4 = 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4 = 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 + 3^4 (3 ways with minimal representation)

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

Cf. A002377.

t=Table[r=PowersRepresentations[n,19,4]; Sort[Tally[19-Count[#,0]&/@r]][[1,2]], {n,800}]; u=Union[t]; c=Complement[Range[Max[u]],u]; If[c=={}, mx=u[[-1]], mx=c[[1]]-1]; Flatten[Table[Position[t,n,1,1],{n,mx}]]

a(10)-a(36) from Alois P. Heinz, Feb 10 2011

A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.

Original entry on oeis.org

1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272, 4295398733, 8590581749

Offset: 1

N. J. A. Sloane

g(n) is the smallest number s such that every natural number is the sum of at most s n-th powers of natural numbers.
It is known (Kubina and Wunderlich, 1990) that g(n) = 2^n + floor((3/2)^n) - 2 for all n <= 471600000. This formula is conjectured to be correct for all n (see A174420).
Mahler showed that there are only finitely many n's for which this formula fails. - Tomohiro Yamada, Sep 23 2017
This sequence (which corresponds to Waring's original conjecture) is much easier to compute than A079611, the problem of finding the minimal s = G(n) for almost all (= sufficienly large) integers. See Wikipedia for a one-line proof that this value for g(n), conjectured by J. A. Euler in 1772, is indeed a lower bound; it is known to be tight if 2^n*frac((3/2)^n) + floor((3/2)^n) <= 2^n, and no counterexample to this inequality is known. - M. F. Hasler, Jun 29 2014

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers (Basic Books 1996) 252-257.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 195.
G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, p. 668.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
S. Pillai, On Waring's Problem, Journal of Indian Math. Soc., 2 (1936), 16-44
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 138.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 239.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 249-250.
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
Edward Waring, Meditationes algebraicae, Cantabrigiae: typis Academicis excudebat J. Archdeacon, 1770.

T. D. Noe, Table of n, a(n) for n = 1..200
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.
Leonard E. Dickson, The Waring Problem and its generalizations, Bulletin of the AMS, 42 (1936) 833-842.
Jeffrey M. Kubina and Marvin C. Wunderlich, Extending Waring's conjecture to 471,600,000, Math. Comp., 55, no. 192 (1990): 815-820.
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math/0412220 [math.NT], 2004.
Feihu Liu and Guoce Xin, On Frobenius Formulas of Power Sequences, arXiv:2210.02722 [math.CO], 2022. See p. 22.
Kurt Mahler, On the fractional parts of the powers of a rational number (II), Mathematika, 4 (1957) 122-124 Math. Rev. 20:33.
Ramin Takloo-Bighash, What about geometry?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 165-185.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, Waring's Problem.
Wikipedia, Waring's Problem.

Cf. A002376, A002377, A079611, A174406, A174420, A297446 (for info on Mathematica functions).

[2^n+Floor((3/2)^n)-2: n in [1..40]]; // Vincenzo Librandi, Aug 15 2015

Maple
```
A002804 := n->2^n+floor( (3/2)^n ) -2;
```

a[n_] := 2^n + Floor[(3/2)^n] - 2; Array[a, 31] (* Robert G. Wilson v, Oct 29 2013 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; a[n_] := 2^n + x[n] - 2; Array[a, 31]  (* Fred Daniel Kline, Jan 11 2018 *)

a(n)=2^n+(3^n>>n)-2 \\ Charles R Greathouse IV, Feb 01 2013

def A002804(n): return (1<>n)-2 # Chai Wah Wu, Jun 25 2024

[2^n+int((3/2)^n)-2 for n in range(1,34)] # Stefano Spezia, Dec 08 2024

A046042 Number of partitions of n into fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9

Offset: 1

Eric W. Weisstein

nonn

In general, the number of partitions of n into perfect s-th powers (s>=1) is asymptotic to (2*Pi)^(-(s+1)/2) * sqrt(s/(s+1)) * k * n^(1/(s+1)-3/2) * exp((s+1)*k*n^(1/(s+1))), where k = (Gamma(1 + 1/s) * Zeta(1 + 1/s) / s)^(s/(s+1)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

			a(33) = 3 because we have [16,16,1], [16,1,1,...,1] (17 1's) and [1,1,...,1] (33 1's).

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Eric Weisstein's World of Mathematics, Partition

Cf. A000583, A002377, A003105.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792, A259793.

a046042 = p $ tail a000583_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, May 18 2015   ~

g:=-1+1/product(1-x^(j^4),j=1..10): gser:=series(g,x=0,105): seq(coeff(gser,x,n),n=1..102); # Emeric Deutsch, Apr 06 2006

g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; gser =
Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}] (* Jean-François Alcover, Oct 29 2012, after Emeric Deutsch *)

G.f.: -1+1/product(1-x^(j^4),j=1..infinity). - Emeric Deutsch, Apr 06 2006
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(1/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(13/10)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
G.f.: Sum_{i>=1} x^(i^4) / Product_{j=1..i} (1 - x^(j^4)). - Ilya Gutkovskiy, May 07 2017

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

A079611 Waring's problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.

Original entry on oeis.org

1, 4, 4, 16, 6, 9, 8, 32, 13, 12, 12, 16, 14, 15, 16, 64, 18, 27, 20, 25

Offset: 1

N. J. A. Sloane, Jan 28 2003

The only certain values are G(1) = 1, G(2) = 4 and G(4) = 16.

See A002804 for the simpler problem of Waring's original conjecture, which does not restrict the bound to "sufficiently large" numbers. - M. F. Hasler, Jun 29 2014

			It is known that every sufficiently large number is the sum of 16 fourth powers, and 16 is the smallest number with this property, so a(4) = G(4) = 16. (The numbers 16^k*31 are not the sum of fewer than 16 fourth powers.)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 395 (shows G(4) >= 16).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

H. Davenport, On Waring's problem for fourth powers, Annals of Mathematics, 40 (1939), 731-747. (Shows that G(4) <= 16.)
Wikipedia, Waring's Problem.
Trevor D. Wooley, On Waring's problem for intermediate powers, arXiv:1602.03221 [math.NT], 2016.

Cf. A002376, A002377, A002804, A174406.

Entry revised Jun 29 2014

A046044 Sum of 13 but no fewer nonzero fourth powers.

Original entry on oeis.org

13, 28, 43, 58, 73, 93, 108, 123, 138, 153, 173, 188, 203, 208, 218, 233, 253, 268, 283, 298, 313, 333, 348, 363, 378, 393, 413, 428, 443, 448, 458, 473, 493, 508, 523, 538, 553, 573, 588, 603, 618, 637, 653, 668, 683, 688, 698, 717, 733, 748, 763, 778, 797

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583, A002377, A047724, A047725.

isSumQ[n_] := Do[pr = PowersRepresentations[n, k, 4]; If[k < 13, If[pr != {} , Return[False]], If[k == 13 && pr != {}, Return[True], Return[False]]], {k, 1, 13}]; Reap[For[n = 1, n <= 800, n++, If[isSumQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2012 *)

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

A046048 Numbers that are the sum of 17 but no fewer nonzero fourth powers.

Original entry on oeis.org

47, 62, 77, 127, 142, 157, 207, 222, 237, 287, 302, 317, 367, 382, 397, 447, 462, 477, 527, 542, 557, 607, 622, 687, 702, 752, 767, 782, 847, 862, 927, 942, 992, 1007, 1022, 1087, 1102, 1167, 1182, 1232, 1247, 1327, 1407, 1487, 1567, 1647, 1727, 1807, 2032

Offset: 1

Eric W. Weisstein

a(65) = 13792 is the last term of this sequence; see A099591 for further references.

			62 is the sum of 17 4th powers and no fewer, so 62 is a term.
63 is the sum of 18 4th powers and no fewer, so 63 is not a term, although it is a term of A099591.

T. D. Noe, Table of n, a(n) for n= 1..65 (full sequence)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583, A002377, A046047, A099591.

lim = 2100; f[n_] := f[n] = (k = 0; While[k++; k <= 17 && PowersRepresentations[n, k, 4] == {}]; k); Select[Range[lim], f[#] == 17 &] (* Jean-François Alcover, Sep 08 2011 *)

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

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)

A047714 Sums of 3 but no fewer nonzero fourth powers.

Original entry on oeis.org

3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

Identical to A003337 for n = 1..87. - Michael S. Branicky, Mar 18 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000583, A002377, A336536, A344187.

Subsequence of A003337.

N:= 3000: # for terms <= N
F1:= {seq(i^4,i=1..floor(N^(1/4)))}: n1:= nops(F1):
F2:= select(`<=`,{seq(seq(F1[i]+F1[j],i=1..j),j=1..nops(F1))},N):
F3:= select(`<=`,{seq(seq(s+t,s=F1),t=F2)},N):
A:= sort(convert(F3 minus (F2 union F1), list)); # Robert Israel, Jul 24 2020

def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
  return sorted(p3-p2-p1)
print(aupto(2400)) # Michael S. Branicky, Mar 18 2021

A002377(a(n)) = 3. - Robert Israel, Jul 24 2020

Equals A003337 - A344187 - A000583. - Sean A. Irvine, May 15 2021

A046046 Sum of 15 but no fewer nonzero fourth powers.

Original entry on oeis.org

15, 30, 45, 60, 75, 95, 110, 125, 140, 155, 175, 190, 205, 220, 235, 240, 255, 270, 285, 300, 315, 335, 350, 365, 380, 395, 415, 430, 445, 460, 475, 480, 495, 510, 525, 540, 555, 575, 590, 605, 620, 639, 655, 670, 685, 700, 719, 720, 735, 750, 765, 780, 799

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000583, A002377, A046045.

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

cp's OEIS Frontend

A185673 Least number k having n representations as the sum of the minimal number of biquadrates A002377(k).

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A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.

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A046042 Number of partitions of n into fourth powers.

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

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A079611 Waring's problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.

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A046044 Sum of 13 but no fewer nonzero fourth powers.

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A046048 Numbers that are the sum of 17 but no fewer nonzero fourth powers.

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