A099591 -id:A099591 - cp's OEIS frontend

A002377 Least number of 4th powers needed to represent n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 1, 2, 3

Offset: 1

N. J. A. Sloane

No terms are greater than 19, see A002804. - Charles R Greathouse IV, Aug 01 2013

Seven values of n need the maximum of 19 fourth powers. These form the arithmetic progression {79, 159, 239, 319, 399, 479, 559} each term being congruent to 79 mod 80. For n < 625 the available fourth powers are congruent to 1 or 16 mod 80, requiring 4*16 + 15*1 to sum to 79. However, 625 = 5^4 is congruent to 65 and 1*65 + 14*1 = 79. So for n > 625 and congruent to 79, only 15 fourth powers are needed to satisfy the mod 80 arithmetic. - Peter Munn, Apr 12 2017

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
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).

T. D. Noe, Table of n, a(n) for n = 1..14000
C. A. Bretschneider, Zerlegung der Zahlen bis 4100 in Biquadrate, J. Reine Angew. Math., 46 (1853), 1-28.
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A002828, A002376, A188462, A374012.

Cf. A046049, A046050, A099591.

Cnt4[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 4]] == 0, k++]; k]; Array[Cnt4, 100] (* T. D. Noe, Apr 01 2011 *)
seq[n_] := Module[{v = Table[0, {n}], s, p}, s = Sum[x^(k^4), {k, 1, n^(1/4)}] + O[x]^(n+1); p=1; For[k=1, k <= 19, k++, p *= s; For[i=1, i <= n, i++, If[v[[i]]==0 && Coefficient[p, x, i] != 0, v[[i]] = k]]]; v];
seq[100] (* Jean-François Alcover, Sep 28 2019, after Andrew Howroyd *)

seq(n)={my(v=vector(n), s=sum(k=1, sqrtint(sqrtint(n)), x^(k^4)) + O(x*x^n), p=1); for(k=1, 19, p*=s; for(i=1, n, if(!v[i] && polcoeff(p,i), v[i]=k))); v} \\ Andrew Howroyd, Jul 06 2018

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002377(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,4,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

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

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

Original entry on oeis.org

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

A186685 Total number of n-digit numbers requiring 19 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 1, 6, 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, 0, 0, 0

Offset: 1

Martin Renner, Feb 25 2011

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

Cf. A046050, A052268, A099591.

Cf. A161905, A186650, A186652, A186654, A186656, A186658, A186660(n) + A186662, A186664, A186666, A186668, A186670, A186672, A186674, A186676, A186678, A186681, A186683, A186684.

PadRight[{0, 1, 6}, 100] (* Paolo Xausa, Jul 26 2024 *)

a(n) = A186684(n) - A186684(n-1).
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) + A186681(n) + A186683(n) + a(n) = A052268(n).
a(n) = 0 for n >= 4. - 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).

A186683 Total number of n-digit numbers requiring 18 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 2, 17, 5, 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, 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) + A186681(n) + a(n) + A186685(n) = A052268(n)

a(n) = 0 for n >= 5. - 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. A046049, A099591.

PadRight[{0, 2, 17, 5}, 100] (* Paolo Xausa, Jul 30 2024 *)

a(n) = A186682(n) - A186682(n-1).

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)

A161884 Smallest k such that n^4 = a_1^4+...+a_k^4 and all a_i are positive integers less than n.

Original entry on oeis.org

16, 6, 16, 5, 6, 6, 16, 6, 5, 7, 6, 6, 6, 5, 16, 6, 6, 6, 5, 6, 7, 6, 6, 5, 6, 6, 6, 6, 5, 5, 16, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 5, 6, 16, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6

Offset: 2

Dmitry Kamenetsky, Jun 21 2009

nonn

It follows from Balasubramanian, Deshouillers, & Dress' result g(4) = 19 that a(n) <= 20. Deshouillers, Hennecart, & Landreau and Deshouillers, Kawada, & Wooley together give an effective proof that G(4) = 16, from which it can be determined by checking the 96 exceptions that a(n) <= 17. Probably a(n) <= 16. [Charles R Greathouse IV, Jul 31 2011]

J.-M. Deshouillers, K. Kawada, and T. D. Wooley, "On sums of sixteen biquadrates", Mem. Soc. Math. Fr. 100 (2005), 120 pp.

Giovanni Resta, Table of n, a(n) for n = 2..250
R. Balasubramanian, J.-M. Deshouillers, and F. Dress, Problème de Waring pour les bicarrés I, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 303 (1986), pp. 85-88.
R. Balasubramanian, J.-M. Deshouillers, and F. Dress, Problème de Waring pour les bicarrés II, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 303 (1986), pp. 161-163.
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.
Jean-Charles Meyrignac, Computing minimal equal sums of like powers
Manfred Scheucher, Sage Script
Eric W. Weisstein, Diophantine Equation 4th Powers

Cf. A161882, A161883, A161885, A099591.

a(n, verbose=0, m=4)={N=n^m; for(k=3, 99, forvec(v=vector(k-1, i, [1, n\sqrtn((k+1-i)*0.99999, m)]), ispower(N-sum(i=1, k-1, v[i]^m), m, &K)&&K>0&&!if(verbose,print1("/*"n" "v"*/"))&&return(k), 1))} \\ M. F. Hasler, Dec 17 2014

a(51)-a(63) from M. F. Hasler, Dec 17 2014

a(64)-a(86) from Giovanni Resta, Aug 17 2015

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).

A374014 Expansion of (Sum_{k>=0} x^(k^4))^16.

Original entry on oeis.org

1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 17, 240, 1680, 7280, 21840, 48048, 80080, 102960, 102960, 80080, 48048, 21840, 7280, 1680, 240, 16, 120, 1680, 10920, 43680, 120120, 240240, 360360, 411840, 360360, 240240, 120120, 43680, 10920, 1680, 120, 0

Offset: 0

Seiichi Manyama, Jun 25 2024

nonn

Number of ways to write n as an ordered sum of 16 fourth powers (A000583).

Seiichi Manyama, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000583, A002377, A014110, A099591.

Cf. A282288, A374016.

my(N=50, x='x+O('x^N)); Vec(sum(k=0, sqrtnint(N, 4), x^k^4)^16)

a(A099591(n)) = 0.

a(n) > 0 for n > 13792.

cp's OEIS Frontend

A002377 Least number of 4th powers needed to represent n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A186685 Total number of n-digit numbers requiring 19 positive biquadrates in their representation as sum of biquadrates.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A186683 Total number of n-digit numbers requiring 18 positive biquadrates in their representation as sum of biquadrates.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A161884 Smallest k such that n^4 = a_1^4+...+a_k^4 and all a_i are positive integers less than n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Extensions

A046047 Sum of 16 but no fewer nonzero fourth powers.

Views

Author

Keywords

Comments

Links