A134341 -id:A134341 - cp's OEIS frontend

A003828 Numbers k such that k^4 is a primitive sum of 3 positive fourth powers.

422481, 2813001, 8707481, 12197457, 16003017, 16430513, 20615673, 44310257, 68711097, 117112081, 145087793, 156646737, 589845921, 638523249, 873822121, 1259768473, 1679142729, 1787882337, 1871713857

Offset: 1

N. J. A. Sloane

There are no further terms up to 1986560000. - Robert Gerbicz, May 17 2009

			The smallest solutions to a^4 + b^4 + c^4 = k^4 are (a,b,c,k) =
95800 217519 414560 422481 (Roger Frye)
673865 1390400 2767624 2813001 (Allan MacLeod)
1705575 5507880 8332208 8707481 (D. J. Bernstein)
5870000 8282543 11289040 12197457 (D. J. Bernstein)
4479031 12552200 14173720 16003017 (D. J. Bernstein)
3642840 7028600 16281009 16430513 (D. J. Bernstein)
2682440 15365639 18796760 20615673 (Noam Elkies)
2164632 31669120 41084175 44310257 (Robert Gerbicz)
10409096 42878560 65932985 68711097 (Robert Gerbicz)
34918520 87865617 106161120 117112081 (Robert Gerbicz)
1841160 121952168 122055375 145087793 (Juergen Rathmann)
27450160 108644015 146627384 156646737 (Juergen Rathmann)
186668000 260052385 582665296 589845921 (Seiji Tomita)
219076465 275156240 630662624 638523249 (Allan MacLeod)
558424440 606710871 769321280 873822121 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
588903336 859396455 1166705840 1259768473 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
50237800 632671960 1670617271 1679142729 (Seiji Tomita)
686398000 1237796960 1662997663 1787882337 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
92622401 1553556440 1593513080 1871713857 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov)
[Example lines revised by _Robert Gerbicz_, May 17 2009]

R. K. Guy, Unsolved Problems in Number Theory, D1.
A. van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 46.

N. Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51:184 (1988), 825-835.
D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d), Math. Comp. 70:233 (2001), 389-394.
Finished distributed computation for solutions up to about 2 billion, Euler Quartic Conjecture (in Russian).
Roger E. Frye, Finding 95800^4 + 217519^4 + 414560^4 = 422481^4 on the Connection Machine, Proceedings of Supercomputing 88, Vol. II: Science and Applications (1988), pp. 106-116.
Index to sequences related to Diophantine equations (4,1,3)

See A078518, A078519, A078520 for values of a, b, c.

Cf. A134341, A175610.

More terms from Robert Gerbicz, Nov 13 2006

Extended by Robert Gerbicz, May 17 2009

A331675 Numbers k such that k^4 = a^4 + b^4 + c^4 + d^4 has at least two positive primitive solutions.

Original entry on oeis.org

31127, 41963, 72899, 154789, 195479, 208471

Offset: 1

Jianing Song, Jan 24 2020

Primitive solutions means gcd(a,b,c,d) = 1.

These are all terms from Jaroslaw Wroblewski link, which gives all positive solutions to k^4 = a^4 + b^4 + c^4 + d^4 where k < 222000, gcd(a,b,c,d) = 1.

			Solutions to k^4 = a^4 + b^4 + c^4 + d^4 = a'^4 + b'^4 + c'^4 + d'^4:
31127: (2260, 4870, 17386, 30335), (2495, 11998, 16430, 30320);
41963: (1100, 17260, 25015, 40234), (8750, 12109, 14470, 41720);
72899: (4555, 44270, 58868, 59330), (9700, 16480, 47618, 69265);
154789: (49586, 55450, 102170, 145615), (66405, 106740, 119760, 121664);
195479: (12970, 43340, 140947, 180520), (25570, 41080, 112822, 189695);
208471: (3903, 46560, 61290, 207950), (91045, 149222, 150550, 168730).

Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000 (Note: (t,m,n) denotes the equation Sum_{i=1..m} (a_i)^t = Sum_{j=1..n} (b_j)^t, where a_i, b_j are positive integers, gcd(a_1, a_2, ..., a_m, b_1, b_2, ..., b_n) = 1.)

Subsequence of A039664 (and thus of A003294).
Other similar sequences:
A023041 (k^3=a^3+b^3+c^3, gcd(a,b,c)=1);
A003828 (k^4=a^4+b^4+c^4, gcd(a,b,c)=1);
A175610 (k^4=a^4+b^4+c^4);
A134341 (k^5=a^5+b^5+c^5+d^5);
A063923 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1);
A063922 (k^5=a^5+b^5+c^5+d^5+e^5);
A331674 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1, at least two solutions).

A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 3, 0, 1, 5, 2, 6, 0, 1, 6, 4, 7, 422481, 0, 1, 7, 3, 4, 353

Offset: 1

Eric Chen, Sep 15 2021

a(26) = T(5,3) is conjectured to be 0, but this has not been proved.
By Fermat's last theorem, T(n,2) = 0 for n > 2.
Euler's sum of powers conjecture is that T(n,k) = 0 for n > k > 1, but this conjecture is not true: T(4,3) = 422481, T(5,4) = 144, there are no known counterexamples for n >= 6.
There are no known 0s for k > 2.
Conjecture: If T(n,k) = 0, then T(r,k) = T(n,s) = T(r,s) = 0 for all r >= n, 2 <= s <= k.

			Table begins:
  n\k |  1   2       3    4   5   6     7     8
  ----+----------------------------------------
   1  |  1   2       3    4   5   6     7     8
   2  |  1   5       3    2   4   3     4     4
   3  |  1   0       6    7   4   3     5     2
   4  |  1   0  422481  353   5   3     9    13
   5  |  1   0       ?  144  72  12    23    14
   6  |  1   0       ?    ?   ?   ?  1141   251
   7  |  1   0       ?    ?   ?   ?   568   102
   8  |  1   0       ?    ?   ?   ?     ?  1409
T(2,5) = 4 because 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 and there is no smaller square that is the sum of 5 positive squares.
T(4,3) = 422481 because 422481^4 = 95800^4 + 217519^4 + 414560^4 and there is no smaller 4th power that is the sum of 3 positive 4th powers.
T(7,7) = 568 because 568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 and there is no smaller 7th power that is the sum of 7 positive 7th powers.

Ed Pegg Jr., Power Sums, Math Games, November 13 2006.
James Waldby, A Table of Fifth Powers equal to Sums of Five Fifth Powers, 2009.
Eric Weisstein's World of Mathematics, Euler's sum of powers conjecture
Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--4th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--6th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--8th Powers
Wikipedia, Euler's sum of powers conjecture

Cf. A007666 (main diagonal), A264764 (subdiagonal for k = n-1).
Cf. A175610 and A003828 (both for a(19) = T(4,3) = 422481).
Cf. A003294 and A039664 (both for a(25) = T(4,4) = 353).
Cf. A134341 (for a(33) = T(5,4) = 144).
Cf. A063922 and A063923 (both for a(41) = T(5,5) = 72).
Cf. A130012, A130022 (these two sequences are not rows of this table, since they require DISTINCT n-th powers, but this table does not have that requirement).

/* return 0 instead of 1 for n=1, and oo loop when T(n, k)=0 */ A347773(p, n, s, m)={ /* Check whether s can be written as sum of n positive p-th powers not larger than m^p. If so, return the base a of the largest term a^p. */ s>n*m^p && return; n==1&&return(ispower(s, p, &n)*n); /* if s and m are not given, s>=n and m are arbitrary. */ !s&&for(m=round(sqrtn(n, p)), 9e9, A347773(p, n, m^p, m-1)&&return(m)); for(a=ceil(sqrtn(s\n, p)), min(sqrtn(max(0, s-n+1), p), m), A347773(p, n-1, s-a^p, a)&&return(a)); } /* after M. F. Hasler in A007666 */ /* Just enter "A347773(n, k)" to get T(n, k) */

T(n,1) = 1.
T(1,k) = k.
T(n,2) = 0 for n > 2.
T(n,n) = A007666(n).
T(n,n-1) = A264764(n).
T(3,k) <= A130012(k).
T(4,k) <= A130022(k).

A376914 Values k for primitive solutions to k^5 + a^5 + b^5 + c^5 = d^5 + e^5 with k >= a >= b >= c > 0 and d >= e >= 0, repetitions allowed.

Original entry on oeis.org

28, 37, 50, 63, 82, 86, 94, 99, 100, 102, 104, 112, 114, 129, 130, 133, 135, 137, 156, 172, 174, 184, 191, 196, 200, 213, 221, 236, 237, 241, 252, 258, 260, 270, 271, 279, 282, 291, 291, 291

Offset: 1

Artur Jasinski, Oct 09 2024

Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
In most of cases d > k.
This case is known in literature as 5.2.4 (see e.g. Eric Weisstein's World of Mathematics).

			28^5 + 20^5 + 10^5 + 4^5 = 29^5 + 3^5 so 28 is a term.
133^5 + 110^5 + 84^5 + 27^5 = 144^5 + 0^5 so 133 is a term.
291^5 + 109^5 + 31^5 + 29^5 = 287^5 + 173^5 and 291^5 + 279^5 + 108^5 + 85^5 = 328^5 + 15^5 and 291^5 + 287^5 + 205^5 + 174^5 = 335^5 + 202^5 so 291 is included three times.

Andrew Howroyd, Solutions for k <= 300, Oct 2024.
Eric Weisstein's World of Mathematics, Diophantine Equation-5th Powers.
Wikipedia, Euler's sum of powers conjecture.

Cf. A063923, A134341, A376289.

aa = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[k^5 + a^5 + b^5 + c^5, 2, 5];If[kk != {}, If[GCD[k,a,b,c,kk[[1]][[1]],kk[[1]][[2]]]==1,Print[{k, a, b, c, kk}]; AppendTo[aa, k]]], {c, 1, b}], {b, 1, a}], {a, 1, k}], {k, 1, 200}], {a, k}]; aa

lista(maxk, mink=1,prfull=0)={for(k=mink, maxk, for(a=1, k, for(b=1, a, for(c=1,b,my(s=k^5+a^5+b^5+c^5);for(d=sqrtnint((s-1)\2,5)+1,  sqrtnint(s,5), my(e); if(ispower(s-d^5,5,&e) && gcd([k,a,b,c,d,e])==1, if(prfull, print([k,a,b,c,d,e]), print1(k, ", ") )) )))))} \\ Andrew Howroyd, Oct 09 2024

a(26)-a(40) from Andrew Howroyd, Oct 09 2024

A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

Original entry on oeis.org

744, 1686, 1921, 2087, 3447, 4097, 6065, 7157, 7864, 8570

Offset: 1

Jianing Song, Jan 24 2020

Primitive solutions means gcd(a,b,c,d,e) = 1.
These are all terms from James Waldby link, which gives all solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 where k < 10000, gcd(a,b,c,d,e) = 1 and at least two of a,b,c,d,e are nonzero.
Note that if nonprimitive solutions were allowed (where at least two of a,b,c,d,e are nonzero), then 144 would be a term because 144^5 = 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 38^5 + 86^5 + 92^5 + 94^5 + 134^5.

			Solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 = a'^5 + b'^5 + c'^5 + d'^5 + e'^5:
744: (100, 210, 414, 629, 651), (14, 95, 545, 586, 644);
1686: (265, 486, 784, 791, 1670), (46, 591, 675, 999, 1655);
1921: (275, 351, 872, 1298, 1855), (95, 771, 1020, 1519, 1756);
2087: (145, 565, 1105, 1462, 1990), (519, 642, 1026, 1480, 1990);
3447: (1212, 1300, 1345, 1699, 3411), (289, 317, 1033, 1682, 3426);
4097: (1281, 2154, 2396, 3462, 3504), (954, 1989, 2127, 2396, 3981);
6065: (3629, 3811, 4070, 4272, 5313), (854, 3160, 3752, 5073, 5196);
7157: (1827, 2186, 4789, 5629, 6376), (930, 2746, 3570, 5109, 6802);
7864: (1093, 2309, 3629, 6137, 7296), (312, 1631, 3418, 3544, 7809);
8570: (1766, 2529, 4086, 5520, 8319), (2101, 2315, 2710, 3960, 8524).

James Waldby, A Table of Fifth Powers equal to a Fifth Power

Subsequence of A063923 (and thus of A063922).
Other similar sequences:
A023041 (k^3=a^3+b^3+c^3, gcd(a,b,c)=1);
A003828 (k^4=a^4+b^4+c^4, gcd(a,b,c)=1);
A175610 (k^4=a^4+b^4+c^4);
A039664 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1);
A003294 (k^4=a^4+b^4+c^4+d^4);
A331675 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1, at least two solutions).
A134341 (k^5=a^5+b^5+c^5+d^5).

A360382 Least integer m whose n-th power can be written as a sum of four distinct positive n-th powers.

Original entry on oeis.org

10, 9, 13, 353, 144

Offset: 1

Zhining Yang, Feb 04 2023

			a(3) = 13 because 13^3 = 1^3 + 5^3 + 7^3 + 12^3 and no smaller cube may be written as the sum of 4 positive distinct cubes.
Terms in this sequence and their representations are:
  10^1 = 1 + 2 + 3 + 4.
  9^2 = 2^2 + 4^2 + 5^2 + 6^2.
  13^3 = 1^3 + 5^3 + 7^3 + 12^3.
  353^4 = 30^4 + 120^4 + 272^4 + 315^4.
  144^5 = 27^5 + 84^5 + 110^5 + 133^5.

Cf. A007666, A039664, A003294, A134341.

n = 5; SelectFirst[
 Range[200], (s =
    IntegerPartitions[#^n, {4, 4}, Range[1, # - 1]^n]^(1/n); (Select[
      s, #[[1]] > #[[2]] > #[[3]] > #[[4]] > 0 &] != {})) &]

def s(n):
    p=[k**n for k in range(360)]
    for k in range(4,360):
        for d in range(k-1,3,-1):
            if 4*p[d]>p[k]:
                cc=p[k]-p[d]
                for c in range(d-1,2,-1):
                    if 3*p[c]>cc:
                        bb=cc-p[c]
                        for b in range(c-1,1,-1):
                           if 2*p[b]>bb:
                               aa=bb-p[b]
                               if aa>0 and aa in p:
                                   a=round(aa**(1/n))
                                   return(n,k,[a,b,c,d])
for n in range(1,6):
    print(s(n))

a(n) = Minimum(m) such that m^n = a^n + b^n + c^n + d^n and 0 < a < b < c < d < m.

cp's OEIS Frontend

A003828 Numbers k such that k^4 is a primitive sum of 3 positive fourth powers.

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A331675 Numbers k such that k^4 = a^4 + b^4 + c^4 + d^4 has at least two positive primitive solutions.

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A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

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A376914 Values k for primitive solutions to k^5 + a^5 + b^5 + c^5 = d^5 + e^5 with k >= a >= b >= c > 0 and d >= e >= 0, repetitions allowed.

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A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

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A360382 Least integer m whose n-th power can be written as a sum of four distinct positive n-th powers.

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