A085318 -id:A085318 - cp's OEIS frontend

A085319 Primes which are the sum of three 5th powers.

3, 307, 487, 9043, 16871, 17293, 17863, 23057, 32359, 32801, 33857, 36739, 40787, 43669, 50599, 59051, 59113, 62417, 65537, 76099, 101267, 104149, 107777, 135893, 160073, 161053, 164419, 249107, 249857, 256609, 259733, 266663, 338909, 340649

Offset: 1

Labos Elemer, Jul 01 2003

nonn

Primes in the sumset {A000584 + A000584 + A000584}. There must be an odd number of odd terms in the sum, either 3 odd terms (as with 3 = 1^5 + 1^5 + 1^5 and 487 = 1^5 + 3^5 + 3^5 and 59051 = 1^5 + 1^5 + 9^5) or two even terms and one odd term (as with 307 = 2^5 + 2^5 + 3^5 and 9043 = 3^5 + 4^5 + 6^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime. - Jonathan Vos Post, Sep 24 2006

			a(1) = 3 = 1^5 + 1^5 + 1^5.
a(2) = 307 = 2^5 + 2^5 + 3^5.
a(3) = 487 = 1^5 + 3^5 + 3^5.
a(4) = 9043 = 3^5 + 4^5 + 6^5.
a(5) = 16871 = 2^5 + 2^5 + 7^5.
a(6) = 17293 = 3^5 + 3^5 + 7^5.

Vladimir Joseph Stephan Orlovsky, Table of n, a(n) for n = 1..1000

Cf. A003348, A007490, A085318.

Cf. A000040, A000584, A003336, A003347.

lim = 10^6; nn = Floor[(lim - 2)^(1/5)]; t = {}; Do[p = i^5 + j^5 + k^5; If[p <= lim && PrimeQ[p], AppendTo[t, p]], {i, nn}, {j, i}, {k, j}]; t = Union[t] (* Vladimir Joseph Stephan Orlovsky and T. D. Noe, Jul 15 2011 *)
Select[Prime[Range[2,30000]],Length[PowersRepresentations[#,3,5]]>0&] (* Harvey P. Dale, Nov 26 2014 *)

A123032 was identical. - T. D. Noe, Jul 15 2011

A283017 Primes which are the sum of three nonzero 6th powers.

Original entry on oeis.org

3, 857, 1459, 4889, 50753, 51481, 66377, 119107, 210961, 262937, 308801, 525017, 531569, 539633, 562691, 766739, 797681, 840241, 1000793, 1046657, 1078507, 1772291, 1864873, 2303003, 2834443, 2986777, 3032641, 3107729, 3365777, 4757609, 4804201, 5135609, 5987593, 7530329, 7534361, 7743529, 8061041

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^6 + y^6 + z^6 where x, y, z > 0.

			3 = 1^6 + 1^6 + 1^6;
857 = 2^6 + 2^6 + 3^6;
1459 = 1^6 + 3^6 + 3^6, etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001014, A003359, A007490, A085317, A085318, A085319, A283018, A283019.

N:= 10^8: # to get all terms <= N
S:= [seq(i^6, i=1..floor(N^(1/6)))]:
S3:= {seq(seq(seq(S[i]+S[j]+S[k],k=1..j),j=1..i),i=1..nops(S))}:
sort(convert(select(t -> t <= N and isprime(t), S3), list)); # Robert Israel, Mar 09 2017

nn = 15; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^6)], # <= nn^6 && PrimeQ[#] &]

list(lim)=my(v=List(),a6,a6b6,t); lim\=1; for(a=1,sqrtnint(lim-2,6), a6=a^6; for(b=1,min(sqrtnint(lim-a6-1,6),a), a6b6=a6+b^6; forstep(c=if(a6b6%2,2,1),min(sqrtnint(lim-a6b6,6),b),2, if(isprime(t=a6b6+c^6), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Mar 09 2017

A283018 Primes which are the sum of three positive 7th powers.

Original entry on oeis.org

3, 257, 82499, 823799, 1119863, 2099467, 4782971, 5063033, 5608699, 6880249, 7160057, 10018571, 10078253, 10094509, 10279937, 10389481, 10823671, 19503683, 20002187, 20388839, 24782969, 31584323, 35850379, 36189869, 37931147, 50614777, 57416131, 62765029, 64845797, 68355029, 71663617, 73028453

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^7 + y^7 + z^7 where x, y, z > 0.

			3 = 1^7 + 1^7 + 1^7;
257 = 1^7 + 2^7 + 2^7;
82499 = 3^7 + 3^7 + 5^7, etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001015, A003370, A007490, A085317, A085318, A085319, A283017, A283019.

N:= 10^9: # to get all terms <= N
Res:= {}:
for x from 1 to floor(N^(1/7)) do
  for y from 1 to min(x, floor((N-x^7)^(1/7))) do
    for z from 1 to min(y, floor((N-x^7-y^7)^(1/7))) do
      p:= x^7 + y^7 + z^7;
      if isprime(p) then Res:= Res union {p} fi
od od od:
sort(convert(Res,list)); # Robert Israel, Feb 26 2017

nn = 14; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^7)], # <= nn^7 && PrimeQ[#] &]

list(lim)=my(v=List(),x7,y7,t,p); for(x=1,sqrtnint(lim\3,7), x7=x^7; for(y=x,sqrtnint((lim-x7)\2,7), y7=y^7; t=x7+y7; forstep(z=y+(x+1)%2,sqrtnint((lim-t)\1,7),2, if(isprime(p=t+z^7), listput(v,p))))); Set(v) \\ Charles R Greathouse IV, Feb 27 2017

A283019 Primes which are the sum of three nonzero 8th powers.

Original entry on oeis.org

3, 6563, 72353, 137633, 787811, 1745153, 7444673, 44726593, 49202147, 61503553, 86093443, 91858243, 100006817, 100072097, 101686177, 107444417, 143046977, 200006561, 214756067, 257412163, 300452323, 430372577, 431661313, 435812033, 447149537, 452523713, 489805633, 530372321, 744340577

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^8 + y^8 + z^8 where x, y, z > 0.

			3 = 1^8 + 1^8 + 1^8;
6563 = 1^8 + 1^8 + 3^8;
72353 = 2^8 + 3^8 + 4^8, etc.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001016, A003381, A006686, A007490, A085317, A085318, A085319, A283017, A283018.

nn = 13; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^8)], # <= nn^8 && PrimeQ[#] &]

list(lim)=my(v=List(),A,B,t); lim\=1; for(a=1,sqrtnint(lim-2,8), A=a^8; for(b=1,min(sqrtnint(lim-A-1,8),a), B=A+b^8; forstep(c=if(B%2,2,1),sqrtnint(lim-B,8),2, if(isprime(t=B+c^8), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

A133740 Primes which are the sum of four positive 4th powers.

Original entry on oeis.org

19, 179, 419, 499, 643, 673, 769, 883, 1153, 1409, 1459, 1889, 2003, 2083, 2131, 2579, 2609, 2659, 2689, 2819, 3169, 3779, 3889, 3907, 4099, 4129, 4259, 4339, 4513, 4723, 4993, 5009, 5059, 5233, 5347, 5443, 5683, 6529, 6659, 6689, 6899, 7219, 7283, 7459

Offset: 1

Jonathan Vos Post, Dec 31 2007

Every positive integer is expressible as a sum of (at most) g(4) = 19 biquadratic numbers (Waring's problem). Davenport (1939) showed that G(4) = 16, meaning that all sufficiently large integers require only 16 biquadratic numbers.

			a(1) = 19 = 2^4 + 1^4 + 1^4 + 1^4 = 16 + 1 + 1 + 1.
a(2) = 179 = 3^4 + 3^4 + 2^4 + 1^4 = 81 + 81 + 16 + 1.
a(3) = 4^4 + 3^4 + 3^4 + 1^4 = 256 + 81 + 81 + 1.

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000040, A000583, A003337, A085318.

Select[Union[ Flatten[Table[ a^4 + b^4 + c^4 + d^4, {a, 1, 10}, {b, 1, a}, {c, 1, b}, {d, 1, c}]]], PrimeQ]

{primes} INTERSECTION {a^4 + b^4 + c^4 + d^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + for a,b,c,d > 0}

A133750 Primes which are the sum of five positive 4th powers.

Original entry on oeis.org

5, 659, 709, 739, 929, 1283, 1409, 1493, 1523, 1877, 1907, 2099, 2179, 2339, 2689, 2803, 3109, 3187, 3299, 3539, 3733, 3923, 4339, 4357, 5009, 5059, 5443, 5683, 5939, 5987, 6053, 6133, 6529, 7219, 7349, 7459, 7699, 7829, 8419, 8609, 8819, 8849, 9043, 9539

Offset: 1

Jonathan Vos Post, Dec 31 2007

Every positive integer is expressible as a sum of (at most) g(4) = 19 biquadratic numbers (Waring's problem). Davenport (1939) showed that G(4) = 16, meaning that all sufficiently large integers require only 16 biquadratic numbers.

			a(1) = 5 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 = 1 + 1 + 1 + 1 + 1.
a(2) = 659 = 5^4 + 2^4 + 2^4 + 1^4 + 1^4 = 625 + 16 + 16 + 1 + 1.
a(3) = 709 = 5^4 + 3^4 + 1^4 + 1^4 + 1^4 = 625 + 81 + 1 + 1 + 1.

Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A000040, A000583, A003337, A085318.

t = Range[9]^4; Select[Union[Plus @@@ Tuples[t, 5]], # < 10^4 && PrimeQ[#] &] (* Giovanni Resta, Jun 20 2016 *)

{primes} INTERSECTION {a^4 + b^4 + c^4 + d^4 + e^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + A000583(e) for a,b,c,d,e > 0}

Data corrected by Giovanni Resta, Jun 20 2016

A140834 Primes that are the sum of at most four nonzero 4th powers.

Original entry on oeis.org

2, 3, 17, 19, 83, 97, 113, 163, 179, 257, 337, 353, 419, 499, 593, 641, 643, 673, 769, 787, 881, 883, 1153, 1297, 1409, 1459, 1553, 1889, 2003, 2083, 2131, 2417, 2579, 2593, 2609, 2657, 2659, 2689, 2819, 3169, 3217, 3697, 3779, 3889, 3907, 4099, 4129, 4177

Offset: 1

Jonathan Vos Post, Jul 18 2008

This sequence was checked by T. D. Noe, who had supplied the b-list for A004833. A037896 is a subset of {Primes that are the sum of at exactly 2 nonzero 4th powers}, itself a subset of A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

Cf. A000040, A000583, A002645, A003294, A004833, A037896, A085318, A122540, A126657, A133740.

A000040 INTERSECTION A004833. {A133740 = Primes that are the sum of at exactly 4 nonzero 4th powers} UNION {A085318 = Primes that are the sum of at exactly 3 nonzero 4th powers} UNION {A002645 = Primes that are the sum of at exactly 2 nonzero 4th powers}.

Missing term 353 inserted by Georg Fischer, May 11 2024

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A085319 Primes which are the sum of three 5th powers.

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A283017 Primes which are the sum of three nonzero 6th powers.

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A283018 Primes which are the sum of three positive 7th powers.

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A283019 Primes which are the sum of three nonzero 8th powers.

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A133740 Primes which are the sum of four positive 4th powers.

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A133750 Primes which are the sum of five positive 4th powers.

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A140834 Primes that are the sum of at most four nonzero 4th powers.

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