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a(n) is the smallest number >= 1 not occurring earlier and not the sum of cubes of two distinct earlier terms.
+0
6
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77
REFERENCES
Mihaly Bencze [Beneze], Smarandache recurrence type sequences, Bulletin of pure and applied sciences, Vol. 16E, No. 2, 1997, pp. 231-236.
F. Smarandache, Properties of numbers, ASU Special Collections, 1973.
MATHEMATICA
A031980 = {1}; Do[ m = Ceiling[(n-1)^(1/3)]; s = Select[ A031980, # <= m &]; ls = Length[s]; sumOfCubes = Union[ Flatten[ Table[ s[[i]]^3 + s[[j]]^3, {i, 1, ls}, {j, i+1, ls}]]]; If[ FreeQ[ sumOfCubes, n], AppendTo[ A031980, n] ], {n, 2, 77}]; A031980 (* Jean-François Alcover, Dec 14 2011 *)
PROG
(Magma) m:=77; a:=[]; a2:={}; for n in [1..m] do p:=1; u:= a2 join { x: x in a }; while p in u do p:=p+1; end while; if p gt m then break; end if; a2:=a2 join { x^3 + p^3: x in a | x^3 + p^3 le m }; Append(~a, p); end for; print a; // Klaus Brockhaus, Jul 16 2008
CROSSREFS
Cf. A024670 (sums of cubes of two distinct positive integers), A001235 (sums of two cubes in more than one way), A141805 (complement).
AUTHOR
J. Castillo (arp(AT)cia-g.com) [Broken email address?]
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Sep 26 2000
Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.
+0
55
2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344
COMMENTS
The sequence is infinite: Fermat proved that numbers expressible as a sum of two positive integral cubes in n different ways exist for any n. Hardy and Wright give a proof in Theorem 412 of An Introduction of Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition).
A001235 gives another definition of "taxicab numbers".
David W. Wilson reports a(6) <= 8230545258248091551205888. [But see next line!]
C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, 2003, show that with high probability, a(6) = 24153319581254312065344.
When negative cubes are allowed, such terms are called "Cabtaxi" numbers, cf. Boyer's web page, Wikipedia or MathWorld. - M. F. Hasler, Feb 05 2013 a(8) <= 50974398750539071400590819921724352 = 58360453256^3 + 370298338396^3 = 7467391974^3 + 370779904362^3 = 39304147071^3 + 370633638081^3 = 109276817387^3 + 367589585749^3 = 208029158236^3 + 347524579016^3 = 224376246192^3 + 341075727804^3 = 234604829494^3 + 336379942682^3 = 288873662876^3 + 299512063576^3. - PoChi Su, May 16 2013 a(9) <= 136897813798023990395783317207361432493888. - PoChi Su, May 17 2013 The preceding bounds are not the best that are presently known.
An upper bound for a(22) was given by C. Boyer (see the C. Boyer link), namely
BTa(22)= 2^12 *3^9 * 5^9 *7^4 *11^3 *13^6 *17^3 *19^3 *31^4 *37^4 *43 *61^3 *73 *79^3 *97^3 *103^3 *109^3 *127^3 *139^3 *157 *181^3 *197^3 *397^3 *457^3 *503^3 *521^3 *607^3 *4261^3.
We also know that (97*491)^3*BTa(22) is an upper bound on a(23), corresponding to the sum x^3+y^3 with
x=2^5 *3^4 *5^3 *7 *11 *13^2 *17 *19^2 *31 *37 *61 *79 *103 *109 *127 *139 *181 *197 *397 *457 *503 *521 *607 *4261 *11836681,
y=2^4 *3^3 *5^3 *7 *11 *13^2 *17 *19 *31 *37 *61 *79 *89 *103 *109 *127 *139 *181 *197 *397 * 457 *503 * 521 *607 *4261 *81929041.
(End)
Conjecture: the number of distinct prime factors of a(n) is strictly increasing as n grows (this is not true if a(7) is equal to the upper bound given above), but never exceeds 2*n. - Sergey Pavlov, Mar 01 2017
REFERENCES
C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
R. K. Guy, Unsolved Problems in Number Theory, D1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition), see Theorem 412.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.
LINKS
D. W. Wilson, Taxicab Numbers (last snapshot available on web.archive.org, as of June 2013).
FORMULA
a(n) > 113*n^3 for n > 1 (a trivial bound based on the number of available cubes; 113 < (1 - 2^(-1/3))^(-3)). - Charles R Greathouse IV, Jun 18 2024
EXAMPLE
Values of {b,c}, a(n) = b^3 + c^3:
n = 1: {1,1}
n = 2: {1, 12}, {9, 10}
n = 3: {167, 436}, {228, 423}, {255, 414}
n = 4: {2421, 19083}, {5436, 18948}, {10200, 18072}, {13322, 16630}
n = 5: {38787, 365757}, {107839, 362753}, {205292, 342952}, {221424, 336588}, {231518, 331954}
n = 6: {582162, 28906206}, {3064173, 28894803}, {8519281, 28657487}, {16218068, 27093208}, {17492496, 26590452}, {18289922, 26224366}. (End)
Numbers that are the sum of 2 positive cubes.
+0
136
2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343
COMMENTS
It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known. [Any further examples are greater than 5 million. - Charles R Greathouse IV, Apr 12 2020] [Any further example is greater than 10^12. - M. F. Hasler, Jan 10 2021] (i) N and N+1 are both the sum of two positive cubes if N=2*(2*n^2 + 4*n + 1)*(4*n^4 + 16*n^3 + 23*n^2 + 14*n + 4), n=1,2,....
(ii) For n >= 2, let N = 16*n^6 - 12*n^4 + 6*n^2 - 2, so N+1 = 16*n^6 - 12*n^4 + 6*n^2 - 1.
Then the identities 16*n^6 - 12*n^4 + 6*n^2 - 2 = (2*n^2 - n - 1)^3 + (2*n^2 + n - 1)^3 16*n^6 - 12*n^4 + 6*n^2 - 1 = (2*n^2)^3 + (2*n^2 - 1)^3 show that N, N+1 are in the sequence. (End)
If n is a term then n*m^3 (m >= 2) is also a term, e.g., 2m^3, 9m^3, 28m^3, and 35m^3 are all terms of the sequence. "Primitive" terms (not of the form n*m^3 with n = some previous term of the sequence and m >= 2) are 2, 9, 28, 35, 65, 91, 126, etc. - Zak Seidov, Oct 12 2011 This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King, May 09 2013 By Fermat's Last Theorem (the special case for exponent 3, proved by Euler, is sufficient), this sequence contains no cubes. - Charles R Greathouse IV, Apr 03 2021
REFERENCES
C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.
LINKS
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
MATHEMATICA
nn = 2*20^3; Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]] (* T. D. Noe, Oct 12 2011 *) With[{upto=2000}, Select[Total/@Tuples[Range[Ceiling[Surd[upto, 3]]]^3, 2], #<=upto&]]//Union (* Harvey P. Dale, Jun 11 2016 *)
PROG
(PARI) cubes=sum(n=1, 11, x^(n^3), O(x^1400)); v = select(x->x, Vec(cubes^2), 1); vector(#v, k, v[k]+1) \\ edited by Michel Marcus, May 08 2017 (PARI) isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1)) \\ M. F. Hasler, Oct 17 2008, improved upon suggestion of Altug Alkan and Michel Marcus, Feb 16 2016 (PARI) T=thueinit('z^3+1); is(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0 \\ Charles R Greathouse IV, Nov 29 2014 (PARI) list(lim)=my(v=List()); lim\=1; for(x=1, sqrtnint(lim-1, 3), my(x3=x^3); for(y=1, min(sqrtnint(lim-x3, 3), x), listput(v, x3+y^3))); Set(v) \\ Charles R Greathouse IV, Jan 11 2022 (Haskell)
a003325 n = a003325_list !! (n-1)
a003325_list = filter c2 [1..] where
c2 x = any (== 1) $ map (a010057 . fromInteger) $
takeWhile (> 0) $ map (x -) $ tail a000578_list
(Python)
from sympy import integer_nthroot
def aupto(lim):
cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]
sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])
return [s for s in sum_cubes if s <= lim]
EXTENSIONS
Error in formula line corrected by Zak Seidov, Jul 23 2009
Numbers that are the sum of two positive cubes in at least three ways (all solutions).
+0
6
87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, 3080802816, 3235261176, 3499524728, 3623721192, 3877315533, 4750893000, 5544709352, 5602516416
REFERENCES
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
R. K. Guy, Unsolved Problems in Number Theory, D1.
LINKS
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine] Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
MATHEMATICA
a=Sort[Flatten@Table[n^3+m^3, {m, 2000}, {n, m-1, 1, -1}]]; f3[l_]:=Module[{t={}}, Do[If[l[[n]]==l[[n+2]], AppendTo[t, l[[n]]]], {n, 1, Length[l]-2}]; t]; f3[a] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
Numbers with at least three 3s in their prime signature.
+0
3
27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000, 1704024, 1809000, 1852200, 1917000, 1971000, 2012472, 2079000, 2133000, 2148552
COMMENTS
In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least three values of i.
The asymptotic density of this sequence is 1 - (1 + s(1) + s(1)^2/2 - s(2)/2) * Product_{p prime} (1-1/p^3+1/p^4) = 0.000018992895371889141564..., where s(k) = Sum_{p prime} ((p-1)/(p^4-p+1))^k. - Amiram Eldar, Jul 22 2024
EXAMPLE
27000 is a term since 27000 = 2^3 * 3^3 * 5^3.
74088 is a term since 74088 = 2^3 * 5^3 * 7^3.
MATHEMATICA
f[n_]:=Count[Last/@FactorInteger[n], 3]>2; Select[Range[10!], f]
PROG
(PARI) is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024
Numbers with at least two 3s in their prime signature.
+0
3
216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000
COMMENTS
In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.
Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).
The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024
EXAMPLE
216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.
MATHEMATICA
f[n_]:=Count[Last/@FactorInteger[n], 3]>1; Select[Range[8!], f]
Numbers that are the sum of two 4th powers in more than one way.
+0
10
635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752
COMMENTS
Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - M. F. Hasler, Feb 21 2015
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D1.
EXAMPLE
a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - M. F. Hasler, Feb 21 2015
MATHEMATICA
Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)
PROG
(PARI) n=4; L=[]; for(b=1, 999, for(a=1, b, t=a^n+b^n; for(c=a+1, sqrtn(t\2, n), ispower(t-c^n, n)||next; print1(t", ")))) \\ M. F. Hasler, Feb 21 2015 (PARI) list(lim)=my(v=List()); for(a=134, sqrtnint(lim, 4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a, 4)+1, min(sqrtnint(lim-a4, 4), a), my(t=a4+b^4); for(c=a+1, sqrtnint(lim, 4), if(ispower(t-c^4, 4), listput(v, t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024
Taxicab numbers that are sandwiched between nonsquarefree numbers.
+0
0
20683, 216125, 327763, 593047, 684019, 842751, 1845649, 2691451, 4505949, 4744376, 5004125, 5772403, 6058747, 7640128, 8029000, 8216000, 8494577, 10702783, 10765603, 10821896, 11859211, 12533824, 13731319, 14916727, 16776487, 18406603, 18617625, 20616463, 22031576, 24480125, 25937576, 27529073
LINKS
Christian Boyer, Les nombres Taxicabs, in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
EXAMPLE
20683 = 13 ⋅ 37 ⋅ 43 (between 20682 = 2 ⋅ 3^3 ⋅ 383 and 20684 = 2^2 ⋅ 5171).
216125 = 5^3 ⋅ 7 ⋅ 13 ⋅ 19 (between 216124 = 2^2 ⋅ 71 ⋅ 761 and 216126 = 2 ⋅ 3^2 ⋅ 12007).
327763 = 31 ⋅ 37 ⋅ 109 (between 327762 = 2 ⋅ 3^2 ⋅ 131 ⋅ 139 and 327764 = 2^2 ⋅ 67 ⋅ 1223).
MATHEMATICA
Select[Import["https://oeis.org/ A001235/b001235.txt", "Table"][[;; , 2]], # < 3*10^7 && Nor @@ SquareFreeQ /@ (# + {-1, 1}) &] (* Amiram Eldar, Apr 25 2024 *)
Taxicab numbers that are sandwiched between semiprimes.
+0
0
4104, 171288, 1728216, 2864288, 2987712, 2991816, 3512808, 3551112, 9016488, 10424232, 12753160, 13825728, 14197248, 36047592, 43450344, 43699392, 49844032, 63057960, 72131904, 75550088, 85188096, 92384712, 107872128, 107919000, 110808000, 117258678, 119824488, 132678000
COMMENTS
All terms are even numbers.
EXAMPLE
4104 = 2^3 * 3^3 * 19 (between 4103 = 11 * 373 and 4105 = 5 * 821).
171288 = 2^3 * 3^3 * 13 * 61 (between 171287 = 157 * 1091 and 171289 = 103 * 1663).
1728216 = 2^3 * 3^5 * 7 * 127 (between 1728215 = 5 * 345643 and 1728217 = 617 * 2801).
MATHEMATICA
Select[Import["https://oeis.org/ A001235/b001235.txt", "Table"][[;; , 2]], # < 1.5*10^8 && PrimeOmega[# + {-1, 1}] == {2, 2} &] (* Amiram Eldar, May 16 2024 *)
Numbers that are the sum of two cubes in at least four ways (primitive solutions).
+0
6
6963472309248, 12625136269928, 21131226514944, 26059452841000, 74213505639000, 95773976104625, 159380205560856, 174396242861568, 300656502205416, 376890885439488, 521932420691227, 573880096718136
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D1.
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