A161883 -id:A161883 - cp's OEIS frontend

A161885 Smallest k such that n^5 = a_1^5+...+a_k^5 and all a_i are positive integers less than n.

32, 26, 19, 18, 14, 12, 9, 11, 9, 13, 6, 12, 8, 10, 9, 8, 10, 10, 9, 10, 10, 7, 6, 9, 7, 9, 8, 9, 6, 9, 6, 8, 7, 7, 6, 8, 7, 8, 7, 7, 7, 9, 8, 9, 7, 8, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 6, 8, 8, 6, 7, 7, 7, 7, 5, 7, 7, 7, 7, 8, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 5, 7, 6, 7, 7, 6, 7

Offset: 2

Dmitry Kamenetsky, Jun 21 2009

nonn

Based on Fermat's Last Theorem: 2 never occurs in this sequence. It is not known whether 3 occurs in this sequence. The first time 4 occurs is for 144^5=27^5+84^5+110^5+133^5.

The first time 5, 6, 7, 8, 9, 10 occur is for n=73, 12, 23, 14, 8, 15; see sequence A252485. - M. F. Hasler, Dec 17 2014

			a(29) = 9 since 29^5 = 5^5 + 9^5 + 10^5 + 10^5 + 10^5 + 15^5 + 15^5 + 17^5 + 28^5, and there is no sum of less than 9 fifth powers equal to 26^5;
a(30) = 6 since 30^5 = 5^5 + 10^5 + 11^5 + 16^5 + 19^5 + 29^5;
a(72) = 5 since 72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5. - _M. F. Hasler_, Dec 17 2014

Giovanni Resta, Table of n, a(n) for n = 2..400
Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers
Eric Weisstein's World of Mathematics, Diophantine Equation 5th Powers

Cf. A161882, A161883, A161884.

a(n,verbose=0,m=5)={N=n^m;for(k=3,32,forvec(v=vector(k-1,i,[1,n\sqrtn(k+1-i,m)]),ispower(N-sum(i=1,k-1,v[i]^m),m,&K)&&K>0&&!(verbose&&print1("/*"n" "v"*/"))&&return(k),1))} \\ M. F. Hasler, Dec 17 2014

a(n)={
    my(N=n^5,k=3);
    while(1,
        my(v=vector(k-1,i,[sqrtnint(max((N-(k-i)*(n-1)^5-1)\i,0),5)+1,sqrtnint((N-i+1)\(k-i+1),5)]));
        forvec(x=v,
            my(s=N-sum(i=1,#x,x[i]^5));
            if(s>0 && ispower(s,5), return(k))
        ,
            1
        );
        k++
    );
}; \\ Charles R Greathouse IV, Dec 18 2014

a(43)-a(83) from M. F. Hasler, Dec 17 2014
a(77) corrected by Charles R Greathouse IV, Dec 19 2014
a(84)-a(100) from Charles R Greathouse IV, Dec 19 2014

A161882 Smallest k such that n^2 = a_1^2 + ... + a_k^2 and all a_i are positive integers less than n.

Original entry on oeis.org

4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 3, 2, 3, 2, 4, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 3, 4, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2

Offset: 2

Dmitry Kamenetsky, Jun 21 2009

nonn

Related to hypotenuse numbers: A161882(A009003(n))=2 for all n.

Jacobi's four-square theorem can be used to show that a(n) <= 4. - Charles R Greathouse IV, Jul 31 2011

			2^2 = 1^2 + 1^2 + 1^2 + 1^2, so a(2)=4.
3^2 = 2^2 + 2^2 + 1^2, so a(3)=3.

Alois P. Heinz, Table of n, a(n) for n = 2..700
Jean-Charles Meyrignac, Computing minimal equal sums of like powers.
Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers.

Cf. A161883, A161884, A161885.

f[n_, k_] := Select[PowersRepresentations[n^2, k, 2], AllTrue[#, 0<#Jean-François Alcover, Oct 03 2020 *)

A161882(n)={vecmin(factor(n)[,1]%4)==1 && return(2);  if(n==1<M. F. Hasler, Dec 17 2014

a(n)=2 iff n is in A009003 (hypotenuse numbers), a(n)=4 iff n is in A000079 (powers of 2), otherwise a(n)=3. - M. F. Hasler, Dec 17 2014

More terms from Alois P. Heinz, Dec 04 2014

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

A252486 Smallest k such that n^6 = a_1^6+...+a_k^6 where all the a_i are positive integers less than n.

Original entry on oeis.org

64, 36, 15, 29, 22, 21, 15, 19, 15, 17, 15, 16, 14, 15, 13, 12, 11, 11, 13, 14, 12, 13, 13, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11, 13, 11, 11, 11, 10, 11, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 9, 11, 10, 11, 11, 11, 9, 10, 11, 11, 11, 11, 10

Offset: 2

M. F. Hasler, Dec 17 2014

nonn

Inspired by Fermat's Last Theorem: 2 never occurs in this sequence.
No n is known for which a(n)<7, according to the MathWorld page. The values 7, 8, 9, 10 and 11 occur first at indices 1141, 251, 54, 39, 18, cf. sequence A252476.
I conjecture that the sequence is bounded by the initial term. Probably even a(3)=36, a(5)=29, a(6)=22 and some more are followed only by smaller terms.
From results on Waring's problem, it is known that all a(n) <= A002804(6) = 73, and a(n) <= 24 for all sufficiently large n. - Robert Israel, Aug 17 2015

Giovanni Resta, Table of n, a(n) for n = 2..200
Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers
Manfred Scheucher, Sage Script
Eric W. Weisstein, Diophantine Equation--6th Powers
Eric W. Weisstein, Waring's Problem

Cf. A161882, A161883, A161884, A161885.

M:= 10^8:
R:= Vector(M, 74, datatype=integer[4]):
for p from 1 to floor(M^(1/6)) do
  p6:= p^6;
  if p > 1 then A[p]:= R[p6] fi;
  R[p6]:= 1;
  for j from p6+1 to M do
    R[j]:= min(R[j], 1+R[j - p6]);
  od
od:
F:= proc(n, k, ub)
   local lb, m, bestyet, res;
   if ub <= 0 then return -1 fi;
   if n <= M then
     if n = 0 then return 0
     elif R[n] > ub then return -1
     else return R[n]
     fi
   fi;
   lb:= floor(n/k^6);
   if lb > ub then return -1 fi;
   bestyet:= ub;
   for m from lb to 0 by -1 do
     res:= procname(n-m*k^6, k-1, bestyet-m);
     if res >= 0 then
       bestyet:= res+m;
     fi
   od:
   return bestyet
end proc:
for n from floor(M^(1/6))+1 to 50 do
   A[n]:= F(n^6, n-1, 73)
od:
seq(A[n], n=2..50); # Robert Israel, Aug 17 2015

a[n_] := Module[{k}, For[k = 7, True, k++, If[IntegerPartitions[n^6, {k}, Range[n-1]^6] != {}, Print[n, " ", k]; Return[k]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 29 2023 *)

a(n,verbose=0,m=6)={N=n^m;for(k=3,64,forvec(v=vector(k-1,i,[1,n\sqrtn(k+1-i,m)]),ispower(N-sum(i=1,k-1,v[i]^m),m,&K)&&K>0&&!(verbose&&print1("/*"n" "v"*/"))&&return(k),1))}

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A252486(n):
    m = n**6
    for k in count(2):
        try:
            next(power_representation(m,6,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

More terms from Manfred Scheucher, Aug 15 2015

a(53)-a(66) from Giovanni Resta, Aug 17 2015

A252487 Smallest k such that n^7 = a_1^7 + ... + a_k^7 and all a_i are positive integers less than n.

Original entry on oeis.org

128, 28, 66, 39, 28, 26, 21, 20, 18, 22, 22, 22, 20, 21, 14, 17, 14, 14, 17, 16, 17, 14, 16, 13, 15, 13, 12, 15, 13, 15, 13, 14, 13, 14, 13, 13, 14, 12, 12, 12, 13, 12, 12, 12, 11, 13, 13, 12, 12, 13, 12, 12, 11, 12, 11, 11, 12, 12, 11, 12, 9, 12, 11, 11, 11

Offset: 2

M. F. Hasler, Dec 17 2014

nonn

Inspired by Fermat's Last Theorem: 2 never occurs in this sequence.
No n is known for which a(n)<7, according to the MathWorld page. The values 7, 8, 9, ... occur first at indices 568, 102, 62, ...
I conjecture that the sequence is bounded by the initial term a(2)=128. Probably even a(4)=66, a(5)=39, a(6)=28 and some more are followed only by smaller terms.
I've uploaded two scripts; one to compute the b-file and one to generate an IP file. For the first script, a parameter kmax can be set to gain a speedup but more memory is used. The other one (which also works with large integers now) should be used in case someone has a good IP-solver. Higher terms might be computable faster with a good IP solver. - Manfred Scheucher, Aug 14 2015
From results on Waring's problem, it is known that all a(n) <= A002804(7) = 143, and a(n) <= 33 for all sufficiently large n. - Robert Israel, Aug 16 2015

Giovanni Resta, Table of n, a(n) for n = 2..200
Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers
Manfred Scheucher, Sage Script for IP-generation
Manfred Scheucher, Sage Script for b-file generation
Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A002804, A161882, A161883, A161884, A161885, A252486.

M:= 10^8:
R:= Vector(M,144, datatype=integer[4]):
for p from 1 to floor(M^(1/7)) do
  p7:= p^7;
  if p > 1 then A[p]:= R[p7] fi;
  R[p7]:= 1;
  for j from p7+1 to M do
    R[j]:= min(R[j],1+R[j - p7]);
  od
od:
F:= proc(n,k,ub)
   local lb, m, bestyet, res;
   if ub <= 0 then return -1 fi;
   if n <= M then
     if n = 0 then return 0
     elif R[n] > ub then return -1
     else return R[n]
     fi
   fi;
   lb:= floor(n/k^7);
   if lb > ub then return -1 fi;
   bestyet:= ub;
   for m from lb to 0 by -1 do
     res:= procname(n-m*k^7, k-1, bestyet-m);
     if res >= 0 then
       bestyet:= res+m;
     fi
   od:
   return bestyet
end proc:
for n from floor(M^(1/7))+1 to 50 do
   A[n]:= F(n^7,n-1,144)
od:
seq(A[n],n=2..50); # Robert Israel, Aug 17 2015

a(n,verbose=0,m=7)={N=n^m;for(k=3,999,forvec(v=vector(k-1,i,[1,n\sqrtn(k+1-i,m)]),ispower(N-sum(i=1,k-1,v[i]^m),m,&K)&&K>0&&!(verbose&&print1("/*"n" "v"*/"))&&return(k),1))}

More terms from Manfred Scheucher, Aug 15 2015

a(50)-a(66) from Giovanni Resta, Aug 17 2015

cp's OEIS Frontend

A161885 Smallest k such that n^5 = a_1^5+...+a_k^5 and all a_i are positive integers less than n.

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A161882 Smallest k such that n^2 = a_1^2 + ... + a_k^2 and all a_i are positive integers less than n.

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

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A252486 Smallest k such that n^6 = a_1^6+...+a_k^6 where all the a_i are positive integers less than n.

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A252487 Smallest k such that n^7 = a_1^7 + ... + a_k^7 and all a_i are positive integers less than n.

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Maple

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