A003336 -id:A003336 - cp's OEIS frontend

A123299 Prime sums of 13 positive 5th powers.

13, 137, 199, 317, 379, 503, 683, 739, 863, 1049, 1129, 1223, 1229, 1409, 1433, 1471, 1613, 1619, 1831, 1949, 1979, 2011, 2221, 2339, 2543, 2549, 2729, 2791, 2909, 2917, 2971, 3089, 3137, 3299, 3307, 3323, 3331, 3361, 3511, 3541, 3659, 3863, 3877, 3931, 4049

Offset: 1

Jonathan Vos Post, Sep 24 2006

			a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 137 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 199 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 317 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294, A123295, A008480.

up = 4100; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)

A000040 INTERSECTION A123299.

a(10)-a(45) from Giovanni Resta, Jun 12 2016

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

A020896 Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.

Original entry on oeis.org

2, 31, 33, 64, 211, 242, 244, 275, 486, 781, 992, 1023, 1025, 1056, 1267, 2048, 2101, 2882, 3093, 3124, 3126, 3157, 3368, 4149, 4651, 6250, 6752, 7533, 7744, 7775, 7777, 7808, 8019, 8800, 9031, 10901, 13682, 15552, 15783, 15961, 16564

Offset: 0

Steven Finch

68101 = (15/2)^5 + (17/2)^5 is believed to be the smallest positive integer k which is the sum of two nonzero fifth powers of rational numbers but not the sum of two nonzero fifth powers of integers.

			31 = 2^5 + (-1)^5.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Cf. A001481, A020897, A003336.

Select[Union[Total/@(Select[Tuples[Range[-8,8],{2}], !MemberQ[#, 0]&]^5)],#>0&]  (* Harvey P. Dale, Apr 03 2011 *)

See Theorem 3.5.6 of J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.

A123300 Prime sums of 14 positive 5th powers.

Original entry on oeis.org

107, 293, 349, 653, 659, 839, 1013, 1019, 1223, 1279, 1409, 1559, 1583, 1621, 1801, 1831, 1949, 2011, 2129, 2153, 2309, 2333, 2339, 2347, 2371, 2551, 2699, 2707, 2731, 2879, 2917, 3083, 3121, 3169, 3191, 3301, 3331, 3449, 3457, 3511, 3541, 3659, 3691, 3761, 3847, 4019, 4027, 4051

Offset: 1

Jonathan Vos Post, Sep 25 2006

			a(1) = 107 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5.
a(2) = 293 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 349 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(4) = 653 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294-A123299.

up = 5000; q = Range[up^(1/5)]^5; a={0}; Do[b = Select[Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 14}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)

A000040 INTERSECTION A123295.

More terms from Harvey P. Dale, Jan 01 2015

4 missing terms from Giovanni Resta, Jun 12 2016

A182198 Primes of form a^2 + b^2 such that a^4 + b^4 is prime.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 53, 73, 89, 137, 149, 157, 181, 257, 269, 281, 293, 313, 349, 373, 397, 401, 409, 421, 461, 541, 557, 577, 593, 661, 709, 733, 757, 769, 773, 797, 853, 937, 953, 1021, 1049, 1069, 1181, 1237, 1277, 1301, 1373, 1429, 1433, 1453, 1489

Offset: 1

Thomas Ordowski, Apr 20 2012

nonn

			13 = 2^2 + 3^2, 2^4 + 3^4 = 97 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002313.
Cf. A003336 (numbers that are the sum of 2 nonzero 4th powers).
Cf. A002645 (quartan primes: primes of the form x^4 + y^4).

nn = 40; t = {}; Do[c = a^2 + b^2; If[c < nn^2 && PrimeQ[c] && PrimeQ[a^4 + b^4], AppendTo[t, c]], {a, nn}, {b, a}]; Sort[t] (* T. D. Noe, Apr 22 2012 *)
Take[#[[1]]^2+#[[2]]^2&/@Select[Tuples[Range[40],2],AllTrue[{#[[1]]^2+ #[[2]]^2, #[[1]]^4+#[[2]]^4},PrimeQ]&]//Union,60] (* Harvey P. Dale, Jun 25 2018 *)

list(lim)=my(v=List(),t);lim\=1;for(x=1,sqrtint(lim),for(y=1, min(sqrtint(lim-x^2),x), if(isprime(t=x^2+y^2)&&isprime(x^4+y^4), listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 22 2012

A343913 Positive integers m such that 2*m^2 - 1 = x^4 + y^4 for some nonnegative integers x and y with |x - y| > 1.

Original entry on oeis.org

71, 347, 1193, 2139, 2709, 17823, 18337, 26057, 32847, 34037, 65793, 87519, 159541, 245573, 383037, 421957, 489731, 520547, 574841, 800589, 1291333, 2010341, 2113003, 2990187, 4528667, 7430553, 8284063, 8402417, 8520567, 9220519, 9865989, 10621507, 11961043, 12335203, 16405581, 17648561, 22224647, 22918853, 24171273

Offset: 1

Zhi-Wei Sun, May 03 2021

nonn

Conjecture: The sequence has infinitely many terms.
Clearly all the terms must be odd and not divisible by 5. Note also that 2*(n^2+n+1)^2 - 1 = n^4 + (n+1)^4.
See also A343917 for a similar conjecture.

			a(1) = 71, and 2*71^2 - 1 = 10^4 + 3^4 with |10 - 3| > 1.
a(53) = 99532937, and 2*99532937^2 - 1 = 19813611095691937 = 11337^4 + 7576^4 with |11337 - 7576| > 1.

Robert Israel, Table of n, a(n) for n = 1..112 (all terms < 10^10; first 53 terms from Zhi-Wei Sun)

Cf. A000290, A000583, A003336, A056220, A343917.

N:= 10^18: # for all terms <= sqrt(N)
R:= {}: count:= 0:
for x from 1 while 2*x^4 < 2*N-1 do
  for y from x+3 by 2 do
    v:= (x^4 + y^4 + 1)/2;
    if v > N then break fi;
    if issqr(v) then
      m:= sqrt(v);
      if not member(m,R) then
         count:= count+1; R:= R union {m};
      fi fi
od od:
sort(convert(R,list)); # Robert Israel, May 04 2021

QQ[n_]:=IntegerQ[n^(1/4)];
n=0;Do[Do[If[QQ[2*m^2-1-(2x)^4]&&Abs[2x-(2*m^2-1-(2x)^4)^(1/4)]>1,n=n+1;Print[n," ",m];Goto[aa]],{x,0,((2m^2-1)^(1/4))/2}];Label[aa],{m,1,25000000}]

A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

Original entry on oeis.org

284, 1388, 2139, 4772, 8556, 8971, 10836, 21163, 28847, 45707, 54507, 71292, 73348, 95127, 101503, 104228, 131388, 136148, 263172, 350076, 638164, 982292, 1532148, 1687828, 1705407, 1958924, 2082188, 2299364, 2360347, 2728379, 3202356, 4042799, 5046771, 5165332, 5235323, 5560627, 7191079, 7740547, 8041364

Offset: 1

Zhi-Wei Sun, May 04 2021

nonn

Conjecture: The sequence has infinitely many terms.
It is easy to see that no term is divisible by 5. In the b-file we list all the 62 terms not exceeding 10^8.
Note that 2*(n^2+3)^2 - 2^4 = (n+1)^4 + (n-1)^4 with (n+1) - (n-1) = 2. This implies that any integer n > 4 can be written as  x + y + 2^(z-1) with x,y,z positive integers such that x^4 + y^4 + (2^z)^4 is twice a square.
See also A343913 for a similar conjecture.

			a(1) = 284, and 2*284^2 - 2^4 = 20^4 + 6^4 with |20-6| > 2.
a(62) = 97077407, and 2*97077407^2 - 2^4 = 18848045899687282 = 11563^4 + 5583^4 with |11563-5583| > 2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..62

Cf. A000290, A000583, A003336, A230121, A230747, A340274, A343862, A343897, A343913.

QQ[n_]:=IntegerQ[n^(1/4)];
n=0;Do[Do[If[QQ[2*m^2-16-x^4]&&(2*m^2-16-x^4)^(1/4)-x>2,n=n+1;Print[n," ",m];Goto[aa]],{x,0,(m^2-8)^(1/4)}];Label[aa],{m,3,8041364}]

A047715 Numbers that are the sum of 4 but no fewer nonzero fourth powers.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

First differs from A003338 at term 64: A003338(64) = 1393 is also a term of A003337, so not a term here. - Michael S. Branicky, Apr 19 2021

Cf. A000583, A002377, A003338 (sum of 4), A003337 (sum of 3), A003336 (sum of 2), A344188, A344187.

limit = 1153
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n,)} if n in qds else set()
  return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
A003338s = set(n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1)
A003337s = set(n for n in range(3, limit+1) if len(findsums(n, 3)) >= 1)
A003336s = set(n for n in range(2, limit+1) if len(findsums(n, 2)) >= 1)
print(sorted(A003338s - A003337s - A003336s - qds)) # Michael S. Branicky, Apr 19 2021

Equals A003338 - A344188 - A344187 - A000583, where "-" denotes "set difference". - Sean A. Irvine, May 15 2021

A123033 Prime sums of 4 positive 5th powers.

Original entry on oeis.org

97, 277, 761, 1511, 1753, 2081, 3221, 3643, 6197, 7517, 7841, 8263, 10067, 10399, 10903, 16903, 25639, 32771, 32833, 33013, 33647, 33889, 35059, 36137, 39019, 40577, 40819, 48563, 49639, 57383, 59083, 59567, 60317, 61129, 62207, 63199, 66383, 66889, 100003

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either one even and 3 odd terms (as with 1^5 + 1^5 + 2^5 + 3^5 and 761 = 2^5 + 3^5 + 3^5 + 3^5) or three even terms and one odd term (as with 97 = 1^5 + 2^5 + 2^5 + 2^5 and 3221 = 2^5 + 2^5 + 2^5 + 5^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 97 = 1^5 + 2^5 + 2^5 + 2^5.
a(2) = 277 = 1^5 + 1^5 + 2^5 + 3^5.
a(3) = 761 = 2^5 + 3^5 + 3^5 + 3^5.
a(7) = 3221 = 2^5 + 2^5 + 2^5 + 5^5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A000584, A003336, A003347, A003349.

up = 10^6; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@Table[e + a, {e, q}], # <= up &]; a = b, {k, 4}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003349.

More terms from Alois P. Heinz, Aug 12 2015

A123034 Prime sums of 5 positive 5th powers.

Original entry on oeis.org

5, 67, 1301, 1543, 2113, 2293, 2777, 3191, 3253, 3347, 3371, 3433, 3613, 4339, 5237, 5417, 5659, 6229, 6737, 7307, 7549, 7873, 8053, 8537, 8803, 9377, 9439, 9619, 9857, 10099, 11177, 11423, 11927, 12743, 15797, 15859, 16811, 17053, 17183, 18679, 18919, 19163

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either 5 odd terms (as with 5 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 and 16811 = 1^5 + 1^5 + 1^5 + 1^5 + 7^5), two even and 3 odd terms (as with 67 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 1301 = 1^5 + 1^5 + 2^5 + 3^5 + 4^5) or four even terms and one odd term (as with 3253 = 2^5 + 2^5 + 2^5 + 2^5 + 5^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 5 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 67 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(3) = 1301 = 1^5 + 1^5 + 2^5 + 3^5 + 4^5.
a(4) = 1543 = 1^5 + 2^5 + 3^5 + 3^5 + 4^5.
a(5) = 2113 = 1^5 + 2^5 + 2^5 + 4^5 + 4^5.
a(6) = 3191 = 1^5 + 1^5 + 2^5 + 2^5 + 5^5.
a(7) = 4339 = 3^5 + 4^5 + 4^5 + 4^5 + 4^5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A000584, A003336, A003347, A003349, A003350.

Take[Select[Union[Total/@Tuples[Range[10]^5,5]],PrimeQ],60] (* Harvey P. Dale, Jul 21 2014 *)

A000040 INTERSECTION A003350.

Corrected and extended by Harvey P. Dale, Jul 21 2014

cp's OEIS Frontend

A123299 Prime sums of 13 positive 5th 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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A020896 Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.

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A123300 Prime sums of 14 positive 5th powers.

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A182198 Primes of form a^2 + b^2 such that a^4 + b^4 is prime.

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A343913 Positive integers m such that 2*m^2 - 1 = x^4 + y^4 for some nonnegative integers x and y with |x - y| > 1.

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A343917 Positive integers m with 2*m^2 - 2^4 = x^4 + y^4 for some nonnegative integers x and y with |x-y| > 2.

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A047715 Numbers that are the sum of 4 but no fewer nonzero fourth powers.

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