A011541 -id:A011541 - cp's OEIS frontend

A133019 Product of n-th prime and n-th prime written backwards.

4, 9, 25, 49, 121, 403, 1207, 1729, 736, 2668, 403, 2701, 574, 1462, 3478, 1855, 5605, 976, 5092, 1207, 2701, 7663, 3154, 8722, 7663, 10201, 31003, 75007, 98209, 35143, 91567, 17161, 100147, 129409, 140209, 22801, 117907, 58843, 127087

Offset: 1

Omar E. Pol, Oct 27 2007

a(8) = 1729 is the second taxicab number, also called the Hardy-Ramanujan number (see A001235, A011541 and A133029).

			a(8) = 1729 because the 8th prime is 19 and 19 written backwards is 91 and 19*91 = 1729.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001235, A004087, A011541, A061205, A067563, A067087, A133029. Prime numbers: A000040.

#*FromDigits[Reverse[IntegerDigits[#]]] & /@ Prime[Range[1, 50]] (* G. C. Greubel, Oct 02 2017 *)
#*IntegerReverse[#]&/@Prime[Range[40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 29 2021 *)

vector(60, n, prime(n)*subst(Polrev(digits(prime(n))), x, 10)) \\ Michel Marcus, Dec 17 2014

a(n) = A000040(n) * A004087(n)

A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

Original entry on oeis.org

1729, 251, 219, 157, 158, 131, 132, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126

Offset: 2

N. J. A. Sloane, Apr 03 2021

nonn

This is r(n,3,2) in Alter's notation.

			a(2) = 1729 = 12^3 + 1^3 = 10^3 + 9^3 (the famous Hardy-Ramanujan number).
a(3) = 251 = 5^3 + 5^3 + 1^3 = 6^3 + 3^3 + 2^3.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 2, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001235, A011541, A230563, A342903, A343077, A343078, A343079, A343081, A343085.

a(n) = n+63 for n >= 9.

A023050 Sum of two coprime cubes in at least three ways.

Original entry on oeis.org

15170835645, 208438080643, 320465258659, 1658465000647, 3290217425101, 3938530307257, 7169838686017, 13112542594333, 24641518275703, 36592635038993, 36848138663889, 41332017729268, 74051580874005

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..41
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Cf. A011541, A003826, A001235, A023051.

A047696 Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.

Original entry on oeis.org

1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, 933528127886302221000

Offset: 1

N. J. A. Sloane

Sometimes called cab-taxi (or cabtaxi) numbers.
For a(10), see the C. Boyer link.
Christian Boyer: After his recent work on Taxicab(6) confirming the number found as an upper bound by Randall Rathbun in 2002, Uwe Hollerbach (USA) confirmed this week that my upper bound constructed in Dec 2006 is really Cabtaxi(10). See his announcement. - Jonathan Vos Post, Jul 08 2008
From PoChi Su, Aug 14 2014: (Start)
An upper bound of a(42) was given by C. Boyer (see the C. Boyer link), denoted by
BCa(42)= 2^9*3^9*5^9*7^7*11^3*13^6*17^3*19^3*29^3*31*37^4*43^4*
        61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*
        193^3*223^3*229^3*307^3*397^3*457^3.
We show that 503^3*BCa(42) is an upper bound of a(43) with an additional sum of x^3+y^3, with
x=2^4*3^3*5^5*7*11*13^2*17*29*37*43*61*67*79*97*101*109*139*163*
  181*193*223*229*307*397*457*2110099,
y=2^3*3^4*5^3*7*11*13^2*17*29*37*41*43*61*67*79*97*101*109*139*163*
  181*193*223*229*307*397*457*176899.
(End)
From PoChi Su, Aug 29 2014: (Start)
An upper bound of a(43) was given by PoChi Su, denoted by
SCa(43)= 2^9*3^9*5^9*7^7*11^3*13^6*17^3*19^3*29^3*31*37^4*43^4*
        61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*
        193^3*223^3*229^3*307^3*397^3*457^3*503^3.
We show that 1307^3*SCa(43) is an upper bound of a(44) with an additional sum of x^3+y^3, with
x=2^3*3^4*5^3*7^2*11*13^2*17*19*23*29*37*43*61*79*101*109*139*163*
  181*193*223*229*307*353*397*457*503*826583,
y=-2^7*3^3*5^3*7^2*11*13^2*17*19^2*29*37*43*61*79*101*109*139*163*
  181*193*223*229*307*397*457*503*58882897.
(End)
From Sergey Pavlov, Feb 18 2017: (Start)
For 1 < n <= 10, each a(n) can be written as the product of not more than n distinct prime powers where one of the factors is a power of 7. For 1 < n <= 9, a(n) can be represented as the difference between two squares, b(n)^2 - c(n)^2, where b(n), c(n) are integers, b(n+1) > b(n), and c(n+1) > c(n):
a(2) = 7 * 13 = 10^2 - 3^2 = 91,
a(3) = 2^3 * 7 * 13 = 33^2 - 19^2,
a(4) = 2^3 * 3^3 * 7^3 * 37 = 1659^2 - 105^2,
a(5) = 3^3 * 7 * 13 * 31 * 79 = 2477^2 - 344^2,
a(6) = 3^3 * 7^4 * 19 * 31 * 37 = 37590^2 - 483^2,
a(7) = 2^3 * 3^3 * 7^4 * 19 * 31 * 37 = 106477^2 - 5929^2,
a(8) = 2^3 * 3^3 * 7^4 * 19 * 23^3 * 31 * 37 = 11736739^2 - 487025^2,
a(9) = 2^3 * 3^3 * 5^3 * 7^4 * 19 * 31 * 37 * 67^3 = 651858879^2 - 3099621^2,
a(10) = 2^3 * 3^3 * 5^3 * 7^4 * 13^3 * 19 * 31 * 37 * 67^3.
(End)

			91 = 6^3 - 5^3 = 4^3 + 3^3 (in two ways).
Cabtaxi(9)=424910390480793000 = 645210^3 + 538680^3 = 649565^3 + 532315^3 = 752409^3 - 101409^3 = 759780^3 - 239190^3 = 773850^3 - 337680^3 = 834820^3 - 539350^3 = 1417050^3 - 1342680^3 = 3179820^3 - 3165750^3 = 5960010^3 - 5956020^3.

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, Section D1.

Daniel J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
Daniel J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
Christian Boyer, New upper bounds on Taxicab and Cabtaxi numbers.
Christian Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Uwe Hollerbach, The tenth cabtaxi number is 933528127886302221000, May 14, 2008.
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
Po-Chi Su, More Upper Bounds on Taxicab and Cabtaxi Numbers, Journal of Integer Sequences, 19 (2016), #16.4.3.
Eric Weisstein's World of Mathematics, Taxicab Numbers
Eric Weisstein's World of Mathematics, Cabtaxi Number
Wikipedia, Cabtaxi number

Cf. A011541, A047697.

a(9) (which was found on Jan 31 2005) from Duncan Moore (Duncan.Moore(AT)nnc.co.uk), Feb 01 2005

A154729 Products of three distinct primes of the form 6*k + 1.

Original entry on oeis.org

1729, 2821, 3367, 3913, 4123, 4921, 5551, 5719, 6097, 6643, 7189, 7657, 8029, 8113, 8827, 8911, 9139, 9331, 9373, 9709, 9919, 10507, 10621, 11137, 11557, 12649, 12901, 13237, 13699, 13741, 14287, 14497, 14539, 14833, 14911, 15067, 15799, 15841

Offset: 1

Omar E. Pol, Jan 18 2009

nonn

a(1) = 1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Equivalently, products of three distinct primes of the form 3*k + 1. - Omar E. Pol, Feb 17 2018

			The first three primes of the form 6*k + 1 are 7, 13 and 19, so a(1) = 7*13*19 = 1729. - _Omar E. Pol_, Feb 17 2018

Felix Fröhlich, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!, Number 1729.

Subsequence of A007304.

Cf. A001235, A002476, A011541, A154716, A154717, A154728.

Module[{nn=40,prs},prs=Select[6*Range[nn]+1,PrimeQ];Take[Times@@@ Subsets[ prs,{3}]//Union,nn]] (* Harvey P. Dale, Feb 17 2018 *)

fct(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ after M. F. Hasler in A027746
is(n) = my(f=fct(n)); if(#f!=3 || f!=vecsort(f, , 8), return(0), for(k=1, #f, if((f[k]-1)/6!=ceil((f[k]-1)/6), return(0)))); 1 \\ Felix Fröhlich, Jul 07 2021

a(5)-a(38) from Donovan Johnson, Jan 28 2009

A003826 Numbers that are the sum of two cubes in at least four ways (primitive solutions).

Original entry on oeis.org

6963472309248, 12625136269928, 21131226514944, 26059452841000, 74213505639000, 95773976104625, 159380205560856, 174396242861568, 300656502205416, 376890885439488, 521932420691227, 573880096718136

Offset: 1

N. J. A. Sloane

nonn

R. K. Guy, Unsolved Problems in Number Theory, D1.

Uwe Hollerbach, Table of n, a(n) for n = 1..664
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]; see also Archive.org backup and local cache [top page only]
E. Rosenstiel et al., The Four Least Solutions ..., Instit. of Mathem. and Its Applic. Bull. Jul 27 (pp. 155-157) 1991
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers
David W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Cf. A011541, A023050, A023051, A001235.

More terms from David W. Wilson, Oct 15 1997

b-file extended by Ray Chandler, Jan 19 2009

A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

Original entry on oeis.org

77976, 223587, 623808, 894348, 1788696, 2105352, 2989441, 4298427, 4672423, 4990464, 5986575, 6036849, 7154784, 8437832, 9747000, 14309568, 16842816, 23915528, 24147396, 24770529, 26745768, 27948375, 34387416, 34634719, 36570744, 37379384, 39923712, 47892600

Offset: 1

Jud McCranie

Observations regarding terms through a(64)=306761364: All are multiples of 7^2, 13^2, and/or 19^2. Other than 2, 3, 5 and 11, their only prime factors are 7, 13, 19, 31, 43, 61, 67, 79, 127, 151, and 181 (each of which exceeds a multiple of 6 by 1). None is a cube or higher power; the ones that are squares are a(7), a(12), a(17), a(19), a(20), a(32), a(33), a(41), a(49), a(55), and a(58). - Jon E. Schoenfield, Oct 08 2006
Many of the terms beyond a(64) have prime factors other than those found in a(1) through a(64); however, each term through a(774) has at most one distinct prime factor p > 5 that does not exceed a multiple of 6 by 1, and p, if such a prime factor exists, has a multiplicity m=3, with only a few exceptions: n=651 and n=713 (where p^m is 11^2), n=346 and n=770 (where p^m is 17^2), n=699 and n=740 (where p^m is 23^2), and n=741 (where p^m is 11^6). - Jon E. Schoenfield, Oct 20 2013
First differs from A145553 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
This sequence is the union of A145553 and A155961.
This sequence is infinite. If n is a member of this sequence, then n^2 = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs. If n^2 = a^3 + b^3 = c^3 + d^3, then (n*k^3)^2 = n^2*k^6 = k^6*(a^3 + b^3) = k^6*(c^3 + d^3) = (a*k^2)^3 + (b*k^2)^3 = (c*k^2)^3 + (d*k^2)^3. It is obvious that if (a, b) and (c, d) are distinct, then (k^2*a, k^2*b), (k^2*c, k^2*d) are also distinct for all nonzero values of k. So if n is in this sequence, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016

			2989441^2 = 1729^3+20748^3 = 15561^3+17290^3, so 2989441 is in the sequence.

Jon E. Schoenfield and Ray Chandler, Table of n, a(n) for n = 1..774

Cf. A050801, A001235, A011541, A145553, A155961.

(* Warning: this script is only a recomputation of the original b-file of 64 terms from Jon E. Schoenfield, and should not be used to extend the data. *)
max = 310000000; cubeFreeParts = {361, 8281, 33124, 159201, 169309, 221725, 565068, 628849, 917427, 1054729, 2370963, 2989441, 4672423, 8968323, 9402967, 9795747, 34634719};
r[x_] := Reduce[0 < y <= z && x^2 == y^3 + z^3, {y, z}, Integers];
okQ[primes_] := Intersection[{2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 67, 79, 127, 139, 151, 181}, primes] == primes;
crop[n_] := Reap[For[m = 1, True, m++, x = n*m^3; If[x > max, Break[]]; If[okQ[FactorInteger[x][[All, 1]]], If[Head[rx = r[x]] === Or, Print["x = ", x, " ", rx]; Sow[x]];]]][[2, 1]];
A051302 = crop /@ cubeFreeParts // Flatten // Sort (* Jean-François Alcover, Jul 02 2017 *)

T=thueinit('x^3+1, 1);
is(n)=my(v=thue(T, n^2)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 10 2016

Definition corrected by Jon E. Schoenfield, Aug 27 2006
More terms from Jon E. Schoenfield, Oct 08 2006
Extended by Ray Chandler, Nov 22 2011

A175362 Number of integer pairs (x,y) satisfying |x|^3 + |y|^3 = n, -n <= x,y <= n.

Original entry on oeis.org

1, 4, 4, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Cube variant of A004018.
Obviously, a(n) must be 4*k, for k >= 0, n > 0. - Altug Alkan, Apr 09 2016
From Robert Israel, Jan 26 2017: (Start)
a(k^3*n) >= a(n) for k >= 1.
a(n) >= 16 for n in A001235.
a(A011541(n)) >= 8*n. (End)

			a(2) = 4 counts (x,y) = (-1,1), (1,1), (-1,-1) and (1,-1).
a(9) = 8 counts (x,y) = (-2,-1), (-2,1), (-1,-2), (-1,2), (1,-2), (1,2), (2,-1) and (2,1).

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001235, A010057, A011541, A025446, A025455, A025464, A025468, A121980.

N:= 200: # to get a(0)..a(N)
G:= (1+2*add(x^(j^3),j=1..floor(N^(1/3))))^2:
seq(coeff(G,x,j),j=0..N); # Robert Israel, Jan 26 2017

a(n) = if(n==0, 1, 4*sum(k=1, sqrtnint(n, 3), ispower(n - k^3, 3))); \\ Daniel Suteu, Aug 16 2021

G.f.: ( 1 + 2 * Sum_{j>=1} x^(j^3) )^2.
a(n^3) = 4 for n > 0. - Altug Alkan, Apr 09 2016
a(n) = 4*Sum_{k=1..floor(n^(1/3))} A010057(n - k^3), for n > 0. - Daniel Suteu, Aug 15 2021

Invalid claim that belonged to A004018 removed by R. J. Mathar, Apr 24 2010

A334013 a(n) is the least integer that is the sum of two nonzero tetrahedral numbers in exactly n ways.

Original entry on oeis.org

2, 140, 102424, 43322844421220

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let Te(k) denote the k-th tetrahedral number, then
a(1) = Te(1) + Te(1);
a(2) = Te(7) + Te(6) = Te(8) + Te(4);
a(3) = Te(82) + Te(34) = Te(83) + Te(27) = Te(84) + Te(7);
a(4) = Te(60500) + Te(33760) = Te(53311) + Te(47682) = Te(63383) + Te(17423) = Te(63495) + Te(15790);

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A010329, A011541, A053587.

a(4) from Giovanni Resta, Apr 13 2020

A085559 Smallest number expressible as the sum of three 4th powers in at least n ways.

Original entry on oeis.org

1, 2673, 811538, 5978882, 137149922, 292965218, 779888018, 5745705602, 49511121842, 49511121842, 281539574498, 281539574498, 7865870969138, 7865870969138, 7865870969138, 7865870969138, 47580188090162

Offset: 1

Stuart Gascoigne (stuart.g(AT)scoigne.com), Jul 04 2003

nonn

The 4th powers used to sum to the numbers in this sequence include some zero terms. Hardy and Wright prove that the sequence is infinite.

			811538 = 29^4 + 17^4 + 12^4 = 28^4 + 21^4 + 7^4 = 27^4 + 23^4 + 4^4.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th Ed (1979) p. 330.

Cf. A011541, A123075.

cp's OEIS Frontend

A133019 Product of n-th prime and n-th prime written backwards.

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A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

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A023050 Sum of two coprime cubes in at least three ways.

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A047696 Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.

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A154729 Products of three distinct primes of the form 6*k + 1.

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A003826 Numbers that are the sum of two cubes in at least four ways (primitive solutions).

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A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

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A175362 Number of integer pairs (x,y) satisfying |x|^3 + |y|^3 = n, -n <= x,y <= n.

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