A039664 -id:A039664 - cp's OEIS frontend

A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.

353, 651, 706, 1059, 1302, 1412, 1765, 1953, 2118, 2471, 2487, 2501, 2604, 2824, 2829, 3177, 3255, 3530, 3723, 3883, 3906, 3973, 4236, 4267, 4333, 4449, 4557, 4589, 4942, 4949, 4974, 5002, 5208, 5281, 5295, 5463, 5491, 5543, 5648, 5658

Offset: 1

N. J. A. Sloane

Sequence gives solutions k to the Diophantine equation A^4 + B^4 + C^4 + D^4 = k^4.
Is this sequence the same as A096739? - David Wasserman, Nov 16 2007
A138760 (numbers k such that k^4 is a sum of 4th powers of four nonzero integers whose sum is k) is a subsequence. - Jonathan Sondow, Apr 06 2008

			353^4 = 30^4 + 120^4 + 272^4 + 315^4.
651^4 = 240^4 + 340^4 + 430^4 + 599^4.
2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4.
2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4.
2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, Curious and interesting numbers, Penguin Books, p. 139.

T. D. Noe, Table of n, a(n) for n = 1..4870 (using Wroblewski's results)
Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236.
Kermit Rose and Simcha Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000.
Index to sequences related to diophantine equations (4,1,4).

Cf. A039664 (subsequence, primitive), A096739.

Cf. also A138760 (subsequence).

fourthPowerSums = {};
Do[a4 = a^4; Do[b4 = b^4; Do[c4 = c^4; Do[d4 = d^4; e4 = a4 + b4 + c4 + d4; e = Sqrt[Sqrt[e4]]; If[IntegerQ[e], AppendTo[fourthPowerSums, e]], {d, c + 1, 9000}], {c, b + 1, 6000}],{b, a + 1, 5000}], {a, 30, 3000}];
Union @ fourthPowerSums (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

Corrected and extended by Don Reble, Jul 07 2007
More terms from David Wasserman, Nov 16 2007
Definition clarified by Jonathan Sondow, Apr 06 2008

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

Original entry on oeis.org

72, 94, 107, 144, 365, 415, 427, 435, 480, 503, 530, 553, 575, 650, 700, 703, 716, 729, 744, 764, 804, 848, 851, 875, 923, 941, 975, 1004, 1006, 1040, 1044, 1235, 1257, 1313, 1327, 1329, 1369, 1392, 1457, 1469, 1504, 1528, 1537, 1575, 1583, 1588, 1596, 1623, 1653, 1685, 1686

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Primitive means a solution for k has gcd(a,b,c,d,e) = 1. [Corrected by Jianing Song, Jan 24 2020]

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Jianing Song, Table of n, a(n) for n = 1..400 (All terms from James Waldby link)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Mathematics of Computation, Vol. 21, No. 99 (Jul., 1967), pp. 446-459. See Table III p. 449.
James Waldby, A Table of Fifth Powers equal to a Fifth Power
Index to sequences related to Diophantine equations (5,1,5)

Cf. A063922.
For cubes: A003072, A023041, A261029.
For fourth powers: A003828, A175610, A039664, A003294.

144 and 1006 inserted and name simplified by Jianing Song, Jan 24 2020

More terms from Jinyuan Wang, Jan 24 2020

A063922 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial solution in nonnegative integers.

Original entry on oeis.org

72, 94, 107, 144, 188, 214, 216, 282, 288, 321, 360, 365, 376, 415, 427, 428, 432, 435, 470, 480, 503, 504, 530, 535, 553, 564, 575, 576, 642, 648, 650, 658, 700, 703, 716, 720, 729, 730, 744, 749, 752, 764, 792, 804, 830, 846, 848, 851, 854, 856, 864, 870

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Any multiple of a term is again a term of this sequence. See A063923 for the primitive solutions. See A007666 for similar solutions for other powers. - M. F. Hasler, Nov 17 2015

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Cf. A063923.

For fourth powers: A003828, A175610, A039664, A003294.

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).

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).

A134820 Primes in the sequence A003294 of certain fourth powers bases.

Original entry on oeis.org

353, 5281, 7703, 9137, 9431, 10939, 11681, 14029, 14489, 17519, 17881, 18077, 18701, 19483, 20719, 21013, 22247, 22961, 24953, 25589, 25913, 26821, 26987, 27893, 28297, 34327, 37273, 39671, 40031, 40129, 42209, 42359, 43397, 43781

Offset: 1

R. J. Mathar, Jan 28 2008

nonn

J. Wroblewski in Puzzle47 of primepuzzles.net.

Cf. A039664, A003294, A096739.

A003294 INTERSECT A000040

A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

1, 625, 157464

Offset: 1

Ilya Gutkovskiy, Dec 30 2021

From Jon E. Schoenfield, Dec 30 2021: (Start)
222000^4 < a(4) < 4891341^4 = lcm(2829, 12259, 16359, 30381)^4 (see A039664, including the Wroblewski link).
10000^5 <= a(5) < 12528^5 = lcm(72, 1044, 1392, 2088, 3132)^5 (see A063923, including the Waldby link; note that, although the terms of A063923 include 72, 144, 1044, 1392, and 2088, whose LCM is only 4176, the primitive solution in which the sum of 5 distinct 5th powers is 144^5 is 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which is not the sum of 5 positive n-th powers).
Conjecture: a(6) = -1. (End)

			For n = 2: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2.
For n = 3: 157464 = 54^3 = 6^3 + 36^3 + 48^3 = 12^3 + 19^3 + 53^3 = 27^3 + 36^3 + 45^3.

Cf. A007666, A025303, A025401, A030052, A039664, A063923, A146756, A230718, A331675.

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

A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.

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A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

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A063922 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial solution in nonnegative integers.

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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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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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A134820 Primes in the sequence A003294 of certain fourth powers bases.

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A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

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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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