A003294 -id:A003294 - cp's OEIS frontend

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

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

A003337 Numbers n which are the sum of 3 nonzero 4th 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

N. J. A. Sloane

Numbers which are in this sequence but not in A047714 must also be the sum of 2 biquadrates, or equal to a fourth power. Among the first 1000 terms of this sequence, this is the case for 4802 = 2*7^4, 57122 = 2*13^4 and 76832 = 2*14^4. - M. F. Hasler, Dec 31 2012
The union of A047714, A336536, and fourth powers of A003294. - Robert Israel, Jul 24 2020
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4.
480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4.
693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A003294, A047714, A336536, A309762, A344188.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

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)
print(aupto(2400)) # Michael S. Branicky, Mar 18 2021

A039664 Numbers k such that k^4 can be written as a sum of four positive 4th powers with no common factor.

Original entry on oeis.org

353, 651, 2487, 2501, 2829, 3723, 3973, 4267, 4333, 4449, 4949, 5281, 5463, 5491, 5543, 5729, 6167, 6609, 6801, 7101, 7209, 7339, 7703, 8373, 8433, 8493, 8517, 8577, 8637, 9137, 9243, 9431, 9519, 9639, 9797, 9877, 10419, 10939, 11681, 11757

Offset: 1

Felice Russo

nonn

T. D. Noe, Table of n, a(n) for n = 1..1003 (from Wroblewski, with duplicate n removed)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A survey of equal sums of like powers, Math. Comp. vol. 21 no. 99, 1967, pp. 446-459, Table 1.
Eric Weisstein's World of Mathematics, Diophantine Equation - 4th powers
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000

Cf. A003294 (nonprimitive solutions allowed), A096739.

Edited by Don Reble, Jul 07 2007

Qualifier "positive" added to the name by Jianing Song, Jan 24 2020

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

A007666 a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

Original entry on oeis.org

1, 5, 6, 353, 72

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The next term a(6) has been claimed to be 1141, but this is incorrect. In fact, 1141^6 is the sum of seven 6th powers. - Jud McCranie, Jun 10 2007
a(7) = 568 and a(8) = 1409. - J. Lowell, Jul 25 2007
a(6) is either 0 (no solution) or greater than 730000 (see the Resta & Meyrignac link, p. 1054). - Jon E. Schoenfield, Jul 22 2017

			1^1 = 1^1.
5^2 = 3^2 + 4^2.
6^3 = 3^3 + 4^3 + 5^3.
353^4 = 30^4 + 120^4 + 272^4 + 315^4.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5.
568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7.
1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 164.

Jean-Charles Meyrignac, FAQ File
Jean-Charles Meyrignac, Resta's Algorithm
Giovanni Resta and Jean-Charles Meyrignac, The smallest solutions to the diophantine equation x^6 + y^6 = a^6 + b^6 + c^6 + d^6 + e^6, Math. Comp. 72 (2003), pp. 1051-1054.

k^n = T(n, 1)^n + ... + T(n, n)^n, where T() is given in A061988.
Examples for n=4 are in A003294.
Examples for n=5 are in A063922.

A007666(n,s,m,p=n)={ /* 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,m,p are not given, s>=n and m are arbitrary and p=n. */ !s&&for(m=round(sqrtn(n,n)),9e9,A007666(n,m^n,m-1,n)&&return(m)); for(a=ceil(sqrtn(s\n,p)),min(sqrtn(s-n+1,p),m),A007666(n-1,s-a^p,a,p)&&return(a));} \\ M. F. Hasler, Nov 17 2015

Name clarified by Dmitry Kamenetsky, Aug 05 2015

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.

A096739 Numbers k such that k^4 can be written as a sum of four distinct positive 4th powers.

Original entry on oeis.org

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, 5729, 5859

Offset: 1

Lekraj Beedassy, May 30 2002

nonn

From David Wasserman, Nov 16 2007: (Start)
Every multiple of a term is a term.
Is this sequence the same as A003294? (End)

			Example solutions:
   353^4 =   30^4 +  120^4 +  272^4 +  315^4;
   706^4 =   60^4 +  240^4 +  544^4 +  630^4;
  1059^4 =   90^4 +  360^4 +  816^4 +  945^4;
  1302^4 =  480^4 +  680^4 +  860^4 + 1198^4;
  1412^4 =  120^4 +  480^4 + 1088^4 + 1260^4;
  3723^4 = 2270^4 + 2345^4 + 2460^4 + 3152^4.

D. Wells, Curious and interesting numbers, Penguin Books, p. 139.

K. Rose and S. Brudno, More about four biquadrates equal one biquadrate, Math. Comp., 27 (1973), 491-494.
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.

Cf. A003294, A176197.

Corrected by Bo Asklund (boa(AT)mensa.se), Nov 05 2004

Corrected and extended by David Wasserman, Nov 16 2007

A297305 Numbers k such that k^4 can be written as a sum of five positive 4th powers.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 31, 35, 40, 45, 50, 55, 60, 62, 65, 70, 75, 80, 85, 89, 90, 93, 95, 100, 103, 105, 110, 115, 120, 124, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 178, 180, 185, 186, 190, 195, 200, 205, 206, 210, 215, 217, 220, 225, 230, 233

Offset: 1

Seiichi Manyama, Mar 16 2018

nonn

If k is in the sequence, then k*m is in the sequence for every positive integer m.

			    5^4 =  2^4 +  2^4 +  3^4 +  4^4 +   4^4 (=       625).
   31^4 = 10^4 + 10^4 + 10^4 + 17^4 +  30^4 (=    923521).
   89^4 = 10^4 + 35^4 + 52^4 + 60^4 +  80^4 (=  62742241).
  103^4 =  4^4 + 15^4 + 50^4 + 50^4 + 100^4 (= 112550881).

Jinyuan Wang, Table of n, a(n) for n = 1..500
Titus Piezas III, Ramanujan and the Quartic Equation 2^4+2^4+3^4+4^4+4^4 = 5^4, 2005.
Jinyuan Wang, All solutions to k^4 = a^4 + b^4 + c^4 + d^4 + e^4 with k < 999.
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.
Index to sequences related to Diophantine equations (4,1,5)

Cf. A000583, A003294, A023042, A301601, A386494.

a(43)-a(57) from Jon E. Schoenfield, Mar 17 2018

A336536 Numbers n that can be written as both the sum of two nonzero fourth powers and the sum of three nonzero fourth powers.

Original entry on oeis.org

4802, 57122, 76832, 260642, 388962, 617057, 913952, 1229312, 1847042, 1957682, 3001250, 3502322, 3748322, 3959297, 4170272, 4626882, 6223392, 6837602, 6959682, 9872912, 11529602, 14623232, 19668992, 21112002, 27691682, 29552672, 31322912, 31505922, 35701250, 40127377, 40302242, 46712801, 48020000, 48355137

Offset: 1

Robert Israel, Jul 24 2020

nonn

The fourth powers are not necessarily distinct.
If n is in the sequence, then so is k^4*n for every k.
The sum of two nonzero fourth powers is never a fourth power (a case of Fermat's last theorem).

			a(3) = 76832 is in the sequence because 76832 = 14^4 + 14^4 = 6^4 + 10^4 + 16^4.
a(6) = 617057 is in the sequence because 617057 = 7^4 + 28^4 = 3^4 + 20^4 + 26^4.

Robert Israel, Table of n, a(n) for n = 1..1072

Cf. A000583, A003294, A047714.

Intersection of A003336 and A003337.

N:= 10^8: # 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):
sort(convert(F3 intersect F2,list));

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)
print(aupto(5*10**7)) # Michael S. Branicky, Mar 18 2021

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).

cp's OEIS Frontend

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

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A003337 Numbers n which are the sum of 3 nonzero 4th powers.

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A039664 Numbers k such that k^4 can be written as a sum of four positive 4th powers with no common factor.

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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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A007666 a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

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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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A096739 Numbers k such that k^4 can be written as a sum of four distinct positive 4th powers.

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A297305 Numbers k such that k^4 can be written as a sum of five positive 4th powers.

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A336536 Numbers n that can be written as both the sum of two nonzero fourth powers and the sum of three nonzero fourth powers.

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