A007490 -id:A007490 - cp's OEIS frontend

A173587 Primes of the form x^3 + 2y^3, with x,y >0.

3, 17, 29, 43, 127, 179, 251, 277, 359, 397, 433, 557, 593, 811, 857, 1051, 1367, 1459, 1583, 1753, 1801, 2017, 2027, 2213, 2251, 2447, 2663, 2689, 2729, 2789, 3221, 3331, 3391, 3457, 3581, 4421, 4519, 4787, 4967, 5653, 6037, 6217, 7109, 7883, 8081

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

Heath-Brown shows that this sequence is infinite.

			a(1) = 1^3+2*1^3 =3, prime. a(2) = 1^3 + 2* 2^3 = 17. a(7) = 1^3+2*r^3 =251.

Zak Seidov, Table of n, a(n) for n = 1..1000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3, Acta Mathematica 186 (2001), pp. 1-84.
H. Iwaniec, Primes represented by quadratic polynomials in two variables, Acta Arithmetica, 24 (1974), 435-459
T. Mitsui, Generalized prime number theorem, Jap.J. Math,26 (1956),1-42

Cf. A007490, A219722.

T:=array(0..5000000): ind:=1: for x from 1 to 1000 do: for y from 1 to 1000 do: z:=x^3 + 2*y^3: if type(z,prime)=true then T[ind] :=z: ind :=ind+1: else fi: od: od: mini:=T[1]: ii:=1: for p from 1 to ind-1 do: for n from 1 to ind-1 do: if T[n] < mini then mini:= T[n]: ii:=n: else fi: od: print(mini): T[ii]:= 999999999999999: ii:=1: mini:=T[1] : od:

formQ[p_] := Reduce[0 < x < p^(1/3) && 0 < y < (p/2)^(1/3) && x^3 + 2 y^3 == p, {x, y}, Integers] =!= False; Select[ Prime[ Range[1100]], formQ] (* Jean-François Alcover, Sep 28 2011 *)

list(lim)=my(v=List(),t); for(y=1,sqrtn(lim\2,3), t=2*y^3; for(x=1,sqrtn(lim-t,3), if(isprime(t+x^3), listput(v,t+x^3)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Sep 28 2011

Converted references to links - R. J. Mathar, Feb 24 2010

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

Original entry on oeis.org

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

A123597 Primes of the form p^3 + q^3 + r^3, where p, q and r are primes.

Original entry on oeis.org

43, 179, 277, 359, 397, 593, 811, 1483, 2017, 2213, 2251, 2447, 2689, 4421, 4519, 4967, 5381, 6271, 7109, 7229, 9181, 9521, 10169, 11897, 12853, 13103, 13841, 14489, 16561, 17107, 20357, 24443, 24677, 25747, 26711, 27917, 30161, 30259, 31193, 31247, 32579, 36161

Offset: 1

Alexander Adamchuk, Nov 14 2006

nonn

a(n) is a subset of A007490(n) = {3, 17, 29, 43, 73, 127, 179, 197, 251, 277, ...}, i.e., primes of the form x^3 + y^3 + z^3.

			a(1) = 43 because 43 = 2^3 + 2^3 + 3^3 is prime and 2^3 + 2^3 + 2^3 = 24 is composite.

Cf. A007490 = Primes of form x^3 + y^3 + z^3.

lst={};Do[Do[Do[p=n^3+m^3+k^3;If[PrimeQ[p]&&PrimeQ[n]&&PrimeQ[m]&&PrimeQ[k],AppendTo[lst,p]],{n,4!}],{m,4!}],{k,4!}];Take[Union[lst],16] (* Vladimir Joseph Stephan Orlovsky, May 23 2009 *)
With[{nn=40},Select[Total/@Tuples[Prime[Range[nn]]^3,3],PrimeQ[#]&&#<= nn^3+ 16&]]//Union (* Harvey P. Dale, Sep 08 2021 *)

A272376 Twin primes both of which are the sum of three positive cubes.

Original entry on oeis.org

2267, 2269, 3527, 3529, 10331, 10333, 14867, 14869, 17207, 17209, 18521, 18523, 18917, 18919, 20231, 20233, 20357, 20359, 25577, 25579, 27791, 27793, 28547, 28549, 31247, 31249, 35279, 35281, 36899, 36901, 40697, 40699, 44279, 44281, 48779, 48781, 51479, 51481

Offset: 1

Carmine Suriano, Apr 28 2016

nonn

			3527 and 3529 are terms since 3527=3^3+5^3+15^3 and 3529=1^3+11^3+13^3.

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

Cf. A001097, A007490, A270225.

cu[n_] := {}!=Quiet@ IntegerPartitions[n,{3},Range[n^(1/3)]^3, 1]; Flatten@ Rest@ Reap@ Do[If[ PrimeQ[p+2] && cu[p] && cu[p+2], Sow[{p, p+2}]], {p, Prime@ Range@ 10000}] (* Giovanni Resta, Apr 28 2016 *)

list(lim)=my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y), if(isprime(t=k+z^3), listput(v, t))))); v=Set(v); for(i=2,#v-1,if(v[i]!=v[i-1]+2 && v[i]!=v[i+1]-2, v[i]=0)); v=Set(v); v[3..#v] \\ Charles R Greathouse IV, Apr 29 2016

A271829 Prime powers p^k such that p^k = x^3 + y^3 + z^3 where x, y, z are positive integers and k > 1, is soluble.

Original entry on oeis.org

81, 729, 2187, 2809, 3481, 5041, 6859, 14641, 15625, 19683, 24389, 26569, 27889, 59049, 63001, 68921, 83521, 148877, 273529, 300763, 332929, 357911, 375769, 413449, 531441, 597529, 619369, 657721, 683929, 704969, 707281, 744769, 776161, 779689, 844561, 877969, 912673

Offset: 1

Altug Alkan, Apr 15 2016

Obviously, this sequence is infinite.
Intersection of A003072 and A025475.
The first terms of this sequence are 3^4, 3^6, 3^7, 53^2, 59^2, 71^2, 19^3, 11^4, 5^6, 3^9, 29^3, 163^2, 167^2, 3^10, ...

			81 is a term because 81 = 3^4 = 3^3 + 3^3 + 3^3.

Cf. A003072, A007490, A025475.

Select[Range[10^6], And[! PrimeQ@ #, PrimePowerQ@ #, Length[PowersRepresentations[#, 3, 3] /. {0, } -> Nothing] > 0] &] (* Michael De Vlieger, Apr 17 2016 *)

list(lim) = my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y),if(isprimepower(k+z^3) && !isprime(k+z^3), listput(v, k+z^3))))); Set(v);

A336451 Primes of form x^3 - (x + 1)^3 + 3z^3 or -x^3 + (x + 1)^3 - 3z^3, with x,z >= 0.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 73, 79, 101, 103, 107, 113, 127, 131, 139, 149, 173, 179, 181, 191, 193, 199, 251, 263, 269, 271, 277, 307, 317, 331, 367, 373, 379, 383, 389, 397, 431, 439, 479, 503, 509, 521, 523, 547, 557, 563, 569, 571

Offset: 1

XU Pingya, Aug 31 2020

nonn

For z <= 10^6, no other prime have this form in the first 105 primes.

			0^3 - 1^3 + 3*2^3 = 23, 23 is a term.
-3^3 + 4^3 - 3*0^3 = -4^3 + 5^3 - 3*2^3 = -52^3 + 53^3 - 3*14^3 = 37, 37 is a term.

Cf. A007490, A221794.

p1 = Select[Prime[Range[105]], IntegerQ[(# - 1)/3] &];
p2 = Select[Prime[Range[105]], IntegerQ[(# + 1)/3] &];
n1 = Length@p1; n2 = Length@p2;
r1 = (p1 - 1)/3; r2 = (p2 + 1)/3;
t = {};
Do[x = (z^3 + r1[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, -x^3 + (x + 1)^3 - 3z^3]], {n, 1,
  n1}, {z, 0, 270}]
Do[x = (z^3 - r2[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, x^3 - (x + 1)^3 + 3z^3]], {n, 1,
  n2}, {z, 0, 170}]
Union@t

cp's OEIS Frontend

A173587 Primes of the form x^3 + 2y^3, with x,y >0.

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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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A123597 Primes of the form p^3 + q^3 + r^3, where p, q and r are primes.

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A272376 Twin primes both of which are the sum of three positive cubes.

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A271829 Prime powers p^k such that p^k = x^3 + y^3 + z^3 where x, y, z are positive integers and k > 1, is soluble.

A336451 Primes of form x^3 - (x + 1)^3 + 3z^3 or -x^3 + (x + 1)^3 - 3z^3, with x,z >= 0.