A086119 -id:A086119 - cp's OEIS frontend

A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

19, 98, 117, 218, 316, 335, 866, 988, 1206, 1304, 1323, 1854, 1946, 2072, 2170, 2189, 2716, 3582, 4570, 4662, 4788, 4886, 4905, 5308, 5402, 5528, 6516, 6734, 6832, 6851, 7254, 9970, 10586, 10836, 11824, 12042, 12140, 12159, 12222, 17530, 17624, 18268

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

To find all differences p^3 - q^3 less than N, it is required that all primes p and q up to sqrt(N/6) be tested.

			117 belongs to the sequence because it can be written as 5^3 - 2^3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A086119, A086121. Also see A045636, A045699.

sumList[x_List, y_List] := (punchline = {}; Do[punchline = Union[punchline, x[[i]] + y], {i, Length[x]}]; punchline); posPart[x_List] := (punchline = {}; Do[If[x[[i]] > 0, punchline = Union[punchline, {x[[i]]}]], {i, Length[x]}]; punchline); posPart[sumList[Prime[Range[10]]^3, - Prime[Range[10]]^3]]
nn=10^5; Union[Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]] (* T. D. Noe, Oct 04 2010 *)
With[{upto=20000},Select[Abs[#[[1]]-#[[2]]]&/@Subsets[Prime[ Range[ Sqrt[ upto/6]]]^3,{2}]//Union,#<=upto&]] (* Harvey P. Dale, Dec 10 2017 *)

Corrected by T. D. Noe, Oct 04 2010

A346917 Numbers that are a sum of the cubes of four primes, not necessarily distinct.

Original entry on oeis.org

32, 51, 70, 89, 108, 149, 168, 187, 206, 266, 285, 304, 367, 383, 386, 402, 405, 424, 484, 500, 503, 522, 601, 620, 702, 718, 721, 740, 819, 838, 936, 1037, 1056, 1154, 1355, 1372, 1374, 1393, 1412, 1472, 1491, 1510, 1589, 1608, 1690, 1706, 1709, 1728, 1807, 1826

Offset: 1

Amiram Eldar, Aug 07 2021

nonn

Roth (1951) proved that the number of terms below x is >> x/log(x)^8.
Ren (2001) proved that this sequence has a positive lower density.
The lower density was proven to be larger than 0.003125 (Ren, 2003), 0.005776 (Liu, 2012), and 0.009664 (Elsholtz and Schlage-Puchta, 2019).

			a(1) = 32 = 2^3 + 2^3 + 2^3 + 2^3.
a(2) = 51 = 2^3 + 2^3 + 2^3 + 3^3.
a(3) = 70 = 2^3 + 2^3 + 3^3 + 3^3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Christian Elsholtz and Jan-Christoph Schlage-Puchta, The density of integers representable as the sum of four prime cubes, arXiv preprint, arXiv:1902.09858 [math.NT], 2019.
Zhixin Liu, Density of the sums of four cubes of primes, Journal of Number Theory, Vol. 132, No. 4 (2012), pp. 735-747.
Xiumin Ren, Density of integers that are the sum of four cubes of primes, Chin. Ann. Math. Ser. B, Vol. 22, No. 2 (2001), pp. 233-242.
Xiumin Ren, Sums of four cubes of primes, J. Number Theory, Vol. 98, No. 1 (2003), pp. 156-171.
K. F. Roth, On Waring's problem for cubes, Proc. London Math. Soc. (2), Vol. 53 (1951), pp. 268-279.

Cf. A030078, A086119, A258865.

seq[max_] := Module[{s = Select[Range[Floor @ Surd[max, 3]], PrimeQ]}, Select[Union[Plus @@@ (Tuples[s, 4]^3)], # <= max &]]; seq[2000]

list(lim)=my(v=List(), P=apply(p->p^3,primes(sqrtnint(lim\=1,3)))); foreach(P,p, foreach(P,q, foreach(P,r, my(s=p+q+r,t); for(i=1,#P, t=s+P[i]; if(t>lim, break); listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Aug 09 2021

from sympy import integer_nthroot, primerange
from itertools import combinations_with_replacement as cwr
def aupto(limit):
    cubes = [p**3 for p in primerange(2, integer_nthroot(limit, 3)[0])]
    return sorted(sum(c) for c in cwr(cubes, 4) if sum(c) <= limit)
print(aupto(2000)) # Michael S. Branicky, Apr 09 2022

A086121 Positive sums or differences of two cubes of primes.

Original entry on oeis.org

16, 19, 35, 54, 98, 117, 133, 152, 218, 250, 316, 335, 351, 370, 468, 686, 866, 988, 1206, 1304, 1323, 1339, 1358, 1456, 1674, 1854, 1946, 2072, 2170, 2189, 2205, 2224, 2322, 2540, 2662, 2716, 3528, 3582, 4394, 4570, 4662, 4788, 4886, 4905, 4921, 4940, 5038

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			117 and 133 each belong to the (set) sequence because can be written as 117 = 5^3 - 2^3 and 133 = 5^3 + 2^3.

Hans Havermann and T. D. Noe, Table of n, a(n) for n = 1..20000.

Cf. A086119, A086120. Also see A045636, A045699.

nn=10^6; td=Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]; ts=Reap[Do[n=Prime[i]^3+Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[nn^(1/3)]}, {j,i}]][[2,1]]; Union[td,ts] (* T. D. Noe, Oct 04 2010 *)
n = 100; Select[Sort@Flatten@ Table[Prime[i]^3 + (-1)^k Prime[j]^3, {i, n}, {j, i}, {k, 2}], 0 < # < (Prime[n] + 2)^3 - Prime[n]^3 &] (* Ray Chandler, Oct 05 2010 *)

Edited by N. J. A. Sloane, Oct 05 2010 to remove a discrepancy between the terms of the sequence and the b-file. The old Mma program and b-file were wrong.

A145767 Numbers n such that n^2 = p^3 + q^3, p, q primes (p<=q).

Original entry on oeis.org

4, 228, 812340, 4935504, 13608420, 14218512, 25354032, 29463060, 48880608, 135516108, 194057640, 223078944, 412662012, 763311948, 764539380, 1409359200, 1998370272, 3408412644, 5397667572, 7650577620, 8224333236, 9401817516

Offset: 1

Zak Seidov, Oct 18 2008

nonn

a(23) > 10^10. [From Donovan Johnson, Nov 18 2009]

			Corresponding values of p and q (p <= q) are: 2, 11, 2137, 8929, 1801, 44111, 6637, 57241, 16931, 151477, 54083, 3889; 2, 37, 8663, 28703, 56999, 48817, 86291, 87959, 133597, 246011, 334717, 367823.

Cf. A086119 Numbers of the form p^3 + q^3, p, q primes.

Cf. A145797, A145798. [From Donovan Johnson, Nov 18 2009]

{n1 = 40000; for(ia= 1, n1,a3 = prime(ia)^3; for ( ib= ia, n1,b3 = prime(ib)^3; c = a3 + b3; if(issquare(c), print([c, a3, b3]))))}

n^2=p^3+q^3, p, q primes (p<=q).

a(13)-a(22) from Donovan Johnson, Nov 18 2009

A145797 Values of p in A145767.

Original entry on oeis.org

2, 11, 2137, 8929, 1801, 44111, 6637, 57241, 16931, 151477, 54083, 3889, 127717, 457511, 571019, 765983, 38557, 1662229, 2437751, 16139, 2305883, 2747449

Offset: 1

Zak Seidov, Oct 19 2008

nonn

A086119, A145767, A145798.

a(13)-a(22) from Donovan Johnson, Nov 18 2009

A145798 Values of q in A145767.

Original entry on oeis.org

2, 37, 8663, 28703, 56999, 48817, 86291, 87959, 133597, 246011, 334717, 367823, 552011, 786697, 735781, 1154017, 1586531, 1915163, 2446777, 3882661, 3811669, 4074743

Offset: 1

Zak Seidov, Oct 19 2008

nonn

A086119, A145767, A145797.

a(13)-a(22) from Donovan Johnson, Nov 18 2009

A294073 Numbers that are the sum of 2 cubes > 1.

Original entry on oeis.org

16, 35, 54, 72, 91, 128, 133, 152, 189, 224, 243, 250, 280, 341, 351, 370, 407, 432, 468, 520, 539, 559, 576, 637, 686, 728, 737, 756, 793, 854, 855, 945, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1339, 1343, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1729, 1736, 1755, 1792, 1843, 1853, 1944, 2000, 2060

Offset: 1

Ilya Gutkovskiy, Feb 07 2018

nonn

			133 is in the sequence because 133 = 2^3 + 5^3.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A003325, A004999, A024670, A085989, A086119.

N:= 3000: # to get all terms <= N
M := floor(N^(1/3)):
sort(convert(select(`<=`, {seq(seq(i^3+j^3, j = 2 .. i), i = 2 .. M)}, N), list)) # Robert Israel, Feb 08 2018

nmax = 2060; f[x_] := Sum[x^k^3, {k, 2, 14}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A320662 Numbers k for which there are numbers 0 < m <= k such that k^3 + m^3 is a square.

Original entry on oeis.org

2, 8, 18, 21, 26, 32, 37, 46, 50, 65, 70, 72, 84, 88, 91, 98, 104, 105, 112, 128, 148, 162, 184, 189, 190, 200, 234, 242, 249, 260, 273, 280, 288, 312, 330, 333, 336, 338, 345, 352, 354, 364, 371, 392, 407, 414, 416, 420

Offset: 1

Marius A. Burtea, Oct 18 2018

nonn

The sequence is infinite since if u is in the sequence then so is u*t^2, t, u >= 1. - Marius A. Burtea and David A. Corneth, Oct 23 2018

For the subsequence k= 8, 18, 32, 50,65, 72, 98, 104, 105,... two or more m exist satisfying the equation. - R. J. Mathar, Jan 22 2025

			8^3 + 4^3 = 512 + 64 = 576 = 24^2, so 8 is part of the sequence.
18^3 + 9^3 = 5832 + 729 = 6561 = 81^2, so 18 is part of the sequence.
91^3 + 65^3 = 753571 + 274625 = 1028196 = 1014^2, so 91 is part of the sequence.
7^3 + 0^3 = 343 + 0 = 343, 7^3 + 1^3 = 343 + 1 = 344, 7^3 + 2^3 = 343 + 8 = 351,7^3 + 4^3 = 343 + 64 = 407, 7^3 + 5^3 = 343 + 125 = 468, 7^3 + 6^3 = 343 + 216 = 559 and 7^3 + 7^3 = 343 + 343 = 686. Numbers 343, 344, 351, 407, 468, 559 and 686 are not squares, so 7 is not part of the sequence.

Cf. A003325, A003997, A004999, A024670, A086119, A282639 (subsequence for coprime m,k), A050801 (bases of the squares).

A320662 := proc(n)
    option remember ;
    local m,k ;
    if n =1 then
        2
    else
        for k from procname(n-1)+1 do
            for m from 1 to k do
                if issqr(k^3+m^3) then
                    return k ;
                end if;
            end do:
        end do:
    end if;
end proc:
seq(A320662(n),n=1..40) ; # R. J. Mathar, Jan 22 2025

Select[Range@ 420, AnyTrue[Range[#1]^3 + #2, IntegerQ@ Sqrt@ # &] & @@ {#, #^3} &] (* Michael De Vlieger, Nov 05 2018 *)

is(n) = for(m=1, n, if(issquare(n^3+m^3), return(1))); 0 \\ Felix Fröhlich, Oct 22 2018

A359448 a(n) is the least number that is the sum of two cubes of primes and is 2^n times an odd number.

Original entry on oeis.org

35, 54, 468, 152, 16, 9056, 81088, 527744, 4532992, 33900032, 268684288, 2148866048, 17185288192, 137439174656, 1099611160576, 8797884612608, 70369850097664, 562950041894912, 4503607335190528, 36028810622664704, 288230406982991872, 2305843633483415552, 18446744212436156416, 147573952867129622528

Offset: 0

Robert Israel, Jan 01 2023

nonn

a(n) is the least member k of A086119 such that A007814(k) = n.
a(n) <= A359447(n) if A359447(n) > 0.
Since p^3 + q^3 = (p+q)*(p^2 - p*q + q^2), except for n=4 we must have A007814(p+q) = n.
There is no analogous sequence for squares, because if p and q are odd primes p^2 + q^2 == 2 (mod 4).

			a(0) = 35 = 2^3 + 3^3 = 2^0 * 35 with 2 and 3 prime and 35 odd.
a(1) = 54 = 3^3 + 3^3 = 2^1 * 27 with 3 and 3 prime and 27 odd.
a(2) = 468 = 5^3 + 7^3 = 2^2 * 117 with 5 and 7 prime and 117 odd.
a(3) = 152 = 3^3 + 5^3 = 2^3 * 19 with 3 and 5 prime and 19 odd.
a(4) = 16 = 2^3 + 2^3 = 2^4 * 1 with 2 and 2 prime and 1 odd.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A007814, A086119, A359447.

f:= proc(n) local p,q,b,t,r;
  r:= infinity;
  for b from 1 by 2 while 2^(3*n-2)*b^3 < r do
    t:= 2^n*b;
    p:= nextprime(t/2);
    while p > 3 do
      p:= prevprime(p);
      q:= t-p;
      if p^3 + q^3 > r then break fi;
      if isprime(q) then r:= p^3 + q^3; break fi;
    od
  od;
    r
end proc:
f(0):= 35: f(4):= 16:
map(f, [$0..30]);

A137632 Sums of 2 cubes of distinct odd primes.

Original entry on oeis.org

152, 370, 468, 1358, 1456, 1674, 2224, 2322, 2540, 3528, 4940, 5038, 5256, 6244, 6886, 6984, 7110, 7202, 8190, 9056, 11772, 12194, 12292, 12510, 13498, 14364, 17080, 19026, 24416, 24514, 24732, 25720, 26586, 29302, 29818, 29916, 30134

Offset: 1

M. F. Hasler, Apr 13 2008

nonn

			3^3 + 5^3 = 152 = a(1).
3^3 + 7^3 = 370 = a(2).
5^3 + 7^3 = 468 = a(3).

A subset of A120398 and A086119. Cf. A138853, A138854.

A137632 := proc(amax) local a,p,q; a := {} ; p := 3 ; while p^3 < amax do q := nextprime(p) ; while p^3+q^3 < amax do a := a union {p^3+q^3} ; q := nextprime(q) ; od: p := nextprime(p) ; od: sort(convert(a,list)) ; end: A137632(80000) ; # R. J. Mathar, May 04 2008

f[upto_]:=Module[{max=Ceiling[Power[upto-27, (3)^-1]],prs}, prs=Prime[Range[2,max]]; Select[Union[Total/@(Subsets[prs,{2}]^3)], #<=upto&]]; f[31000] (* Harvey P. Dale, Apr 20 2011 *)

More terms from R. J. Mathar, Apr 13 2008, May 04 2008

cp's OEIS Frontend

A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

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A346917 Numbers that are a sum of the cubes of four primes, not necessarily distinct.

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A086121 Positive sums or differences of two cubes of primes.

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A145767 Numbers n such that n^2 = p^3 + q^3, p, q primes (p<=q).

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A145797 Values of p in A145767.

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A145798 Values of q in A145767.

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A294073 Numbers that are the sum of 2 cubes > 1.

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A320662 Numbers k for which there are numbers 0 < m <= k such that k^3 + m^3 is a square.

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A359448 a(n) is the least number that is the sum of two cubes of primes and is 2^n times an odd number.

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