A008917 -id:A008917 - cp's OEIS frontend

A001235 Taxi-cab numbers: sums of 2 cubes in more than 1 way.

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977

Offset: 1

N. J. A. Sloane

From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"
A011541 gives another version of "taxicab numbers".
If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - Altug Alkan, May 09 2016

			4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.

R. K. Guy, Unsolved Problems in Number Theory, Section D1.
G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
Ya. I. Perelman, Algebra can be fun, pp. 142-143.
H. W. Richmond, On integers which satisfy the equation t^3 +- x^3 +- y^3 +- z^3, Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.

Shahar Amitai, Table of n, a(n) for n = 1..30000 (terms a(1)-a(4724) from T. D. Noe, terms a(4725)-a(10000) from Zak Seidov).
Shahar Amitai, Python code to generate all taxicab numbers up to N.
J. Charles-É, Recreomath, Ramanujan's Number.
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Henk Koppelaar, Peyman Nasehpour, and Maarten Looijen, Symmetry between Series if Entangled by Sums, Preprints.org, 2024.
Istanbul Bilgi University, Ramanujan and Hardy's Taxi
Christopher Lane, The First ten Ta(2) and their double distinct cubic sums representations, Find Ramanujan's Taxi Number using JavaScript. [WayBack Machine copy]
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
J. Loy, The Hardy-Ramanujan Number.
Mia Muessig, Julia code for finding general taxicab numbers
Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Diophantine Equation 3rd Powers
Eric Weisstein's World of Mathematics, Taxicab Number
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Subsequence of A003325.
Cf. A007692, A008917, A011541, A018786, A018850 (primitive solutions), A051347 (allows negatives), A343708, A360619.
Solutions in greater numbers of ways:
(>2): A018787 (A003825 for primitive, A023050 for coprime),
(>3): A023051 (A003826 for primitive),
(>4): A051167 (A155057 for primitive).

Select[Range[750000],Length[PowersRepresentations[#,2,3]]>1&] (* Harvey P. Dale, Nov 25 2014, with correction by Zak Seidov, Jul 13 2015 *)

is(n)=my(t);for(k=ceil((n/2)^(1/3)),(n-.4)^(1/3),if(ispower(n-k^3,3),if(t,return(1),t=1)));0 \\ Charles R Greathouse IV, Jul 15 2011

T=thueinit(x^3+1,1);
is(n)=my(v=thue(T,n)); sum(i=1,#v,v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 09 2016

A024796 Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k.

Original entry on oeis.org

27, 33, 38, 41, 51, 54, 57, 59, 62, 66, 69, 74, 75, 77, 81, 83, 86, 89, 90, 94, 98, 99, 101, 102, 105, 107, 108, 110, 113, 114, 117, 118, 121, 122, 123, 125, 126, 129, 131, 132, 134, 137, 138, 139, 141, 146, 147, 149, 150, 152, 153, 154, 155, 158, 161, 162, 164, 165, 166, 170

Offset: 1

Clark Kimberling

nonn

a(n) multiplied by (h^2)/(8*m*a^2) is the n-th energy level exhibiting accidental degeneracy, for a quantum mechanical particle of mass m in a cubic box of side length a (h is Planck's constant). - A. Timothy Royappa, Feb 12 2019

Robert Price, Table of n, a(n) for n = 1..8057

Cf. A000408, A007692, A008917, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025367, A025427.

okQ[n_]:= Length[Select[PowersRepresentations[n, 3, 2], !MemberQ[#, 0] &]] > 1; (* Jinyuan Wang, Feb 12 2019 *)

is(n)=if(n<27, return(0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(1)))); 0 \\ Charles R Greathouse IV, Aug 05 2024

{n: A025427(n) > 1 }. - R. J. Mathar, Aug 05 2022

A025456 Number of partitions of n into 3 positive cubes.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

David W. Wilson

nonn

If A025455(n) > 0 then a(n + k^3) > 0 for k>0; a(A119977(n))>0; a(A003072(n))>0. - Reinhard Zumkeller, Jun 03 2006
a(A057904(n))=0; a(A003072(n))>0; a(A025395(n))=1; a(A008917(n))>1; a(A025396(n))=2. - Reinhard Zumkeller, Apr 23 2009
The first term > 1 is a(251) = 2. - Michel Marcus, Apr 23 2019

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Least inverses are A025418.

Cf. A025455, A003108, A003072 (1 or more ways), A008917 (two or more ways), A025395-A025398.

A025456 := proc(n)
    local a,x,y,zcu ;
    a := 0 ;
    for x from 1 do
        if 3*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+2*y^3 > n then
                break;
            end if;
            zcu := n-x^3-y^3 ;
            if isA000578(zcu) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

a[n_] := Count[ PowersRepresentations[n, 3, 3], pr_List /; FreeQ[pr, 0]]; Table[a[n], {n, 0, 107}] (* Jean-François Alcover, Oct 31 2012 *)

a(n)=sum(a=sqrtnint(n\3,3),sqrtnint(n,3),sum(b=1,a,my(C=n-a^3-b^3,c);ispower(C,3,&c)&&0Charles R Greathouse IV, Jun 26 2013

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Second offset from Michel Marcus, Apr 23 2019

A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

Original entry on oeis.org

5104, 9729, 12104, 12221, 12384, 13896, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687

Offset: 1

David W. Wilson

nonn

			a(1) = A230477(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3. - _Jonathan Sondow_, Oct 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A008917, A018787, A025331, A025397, A025402, A025407, A230477, A343968, A344239.

Select[ Range[ 50000], 2 < Length @ Cases[ PowersRepresentations[#, 3, 3], {?Positive, ?Positive, A008917%20by%20_Jonathan%20Sondow">?Positive}] &] (* adapted from Alcover's program for A008917 by _Jonathan Sondow, Oct 24 2013 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1,k,for(b=a,k,for(c=b,k,if(a^3+b^3+c^3==n,d++))));d
n=3;while(n<50000,if(is(n)>=3,print1(n,", "));n++) \\ Derek Orr, Aug 27 2015

A008917(n) < a(n) <= A025397(n). - Jonathan Sondow, Oct 24 2013

{n: A025456(n) >= 3}. - R. J. Mathar, Jun 15 2018

A025396 Numbers that are the sum of 3 positive cubes in exactly 2 ways.

Original entry on oeis.org

251, 1009, 1366, 1457, 1459, 1520, 1730, 1737, 1756, 1763, 1793, 1854, 1945, 2008, 2072, 2241, 2414, 2456, 2458, 2729, 2736, 3060, 3391, 3457, 3592, 3599, 3655, 3745, 3926, 4105, 4112, 4131, 4168, 4229, 4320, 4376, 4402, 4437, 4447, 4473, 4528, 4616

Offset: 1

David W. Wilson

nonn

Subset of A008917; A025397 gives examples of numbers which are in A008917 but not here. - R. J. Mathar, May 28 2008
A025456(a(n)) = 2. - Reinhard Zumkeller, Apr 23 2009
Superset of A024974 . - Christian N. K. Anderson, Apr 11 2013

			a(1) = 251 = 1^3+5^3+5^3 = 2^3+3^3+6^3. - _Christian N. K. Anderson_, Apr 11 2013

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Decomposition of the first 10000 terms into the sets of three cubes

Cf. A008917, A025322, A025395, A025397, A025404, A343708, A344192.

Select[Range[5000], Length[DeleteCases[PowersRepresentations[#,3,3], ?(MemberQ[#,0]&)]] == 2&] (* _Harvey P. Dale, Jan 18 2012 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1,k,for(b=a,k,for(c=b,k,if(a^3+b^3+c^3==n,d++))));d
n=3;while(n<5000,if(is(n)==2,print1(n,", "));n++) \\ Derek Orr, Aug 27 2015

A309762 Numbers that are the sum of 3 nonzero 4th powers in more than one way.

Original entry on oeis.org

2673, 6578, 16562, 28593, 35378, 42768, 43218, 54977, 94178, 105248, 106353, 122018, 134162, 137633, 149058, 171138, 177042, 178737, 181202, 195122, 195858, 198497, 216513, 234273, 235298, 235553, 264113, 264992, 300833, 318402, 318882, 324818, 334802, 346673

Offset: 1

Ilya Gutkovskiy, Aug 15 2019

nonn

			2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, so 2673 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..1616
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A003337, A008917, A018786, A024796, A193243, A309763, A344192, A344239, A345010.

N:= 10^6: # for terms <= N
V:= Vector(N,datatype=integer[4]):
for a from 1 while a^4 <= N do
  for b from 1 to a while a^4+b^4 <= N do
    for c from 1 to b do
      v:= a^4+b^4+c^4;
      if v > N then break fi;
      V[v]:= V[v]+1
od od od:
select(i -> V[i]>1, [$1..N]); # Robert Israel, Aug 19 2019

Select[Range@350000, Length@Select[PowersRepresentations[#, 3, 4], ! MemberQ[#, 0] &] > 1 &]

A024974 Numbers that are the sum of 3 distinct positive cubes in 2 or more ways.

Original entry on oeis.org

1009, 1366, 1457, 1520, 1737, 1756, 1793, 1854, 1945, 2072, 2241, 2456, 2736, 3060, 3592, 3599, 3745, 3926, 4105, 4131, 4168, 4229, 4320, 4376, 4437, 4447, 4473, 4616, 4733, 4922, 5104, 5130, 5435, 5472, 5643, 5706, 5825, 5832, 6183, 6301, 6642, 6848

Offset: 1

Clark Kimberling

nonn

			a(1) = 1009 = 1^3+2^3+10^3 = 4^3+6^3+9^3. - _Christian N. K. Anderson_, Apr 11 2013

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Cube decompositions of the first 10000 terms

Cf. A025400 (exactly 2 ways), A008917 (not necessarily distinct)

{n: A025469(n) >= 2}. - R. J. Mathar, Jun 15 2018

A025397 Numbers that are the sum of 3 positive cubes in exactly 3 ways.

Original entry on oeis.org

5104, 9729, 12104, 12221, 12384, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687, 43408, 45144

Offset: 1

David W. Wilson

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A008917, A025323, A025396, A025398, A025405, A025456, A343969, A344240, A344804.

N:= 10^5: # to get all terms <= N
Reps:= Matrix(N,3,(i,j) -> {}):
for i from 1 to floor(N^(1/3)) do
  Reps[i^3,1]:= {[i]}
od:
for j from 2 to 3 do
for i from 1 to floor(N^(1/3)) do
  for x from i^3+1 to N do
    Reps[x,j]:= Reps[x,j] union
      map(t -> if t[-1] <= i then [op(t),i] fi, Reps[x-i^3,j-1]);
  od
od
od:
select(t -> nops(Reps[t,3])=3, [$1..N]); # Robert Israel, Aug 28 2015

Reap[ For[ n = 1, n <= 50000, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 3, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1, k, for(b=a, k, for(c=b, k, if(a^3+b^3+c^3==n, d++)))); d
n=3; while(n<50000, if(is(n)==3, print1(n, ", ")); n++) \\ Derek Orr, Aug 27 2015

n such that A025456(n) = 3. - Robert Israel, Aug 28 2015

A025406 Numbers that are the sum of 4 positive cubes in 2 or more ways.

Original entry on oeis.org

219, 252, 259, 278, 315, 376, 467, 522, 594, 702, 758, 763, 765, 802, 809, 819, 856, 864, 945, 980, 1010, 1017, 1036, 1043, 1073, 1078, 1081, 1118, 1134, 1160, 1225, 1251, 1352, 1367, 1368, 1374, 1375, 1393, 1397, 1423, 1430, 1458, 1460, 1465, 1467, 1484

Offset: 1

David W. Wilson

nonn

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

Cf. A003327, A008917, A025367, A025404, A025407, A025457, A309763, A343702.

N:= 2000: # for terms <= N
S2:= {}: S1:= {}:
for x from 1 while x^3 < N do
for y from 1 to x while x^3 + y^3 < N do
  for z from 1 to y while x^3 + y^3 + z^3 < N do
    for w from 1 to z do
    v:= x^3 + y^3 + z^3 + w^3;
    if v > N then break fi;
    if member(v,S1) then S2:= S2 union {v}
    else S1:= S1 union {v}
    fi
od od od od:
sort(convert(S2,list)); # Robert Israel, Feb 24 2021

{n: A025457(n) >= 2}. - R. J. Mathar, Jun 15 2018

A001239 Numbers that are the sum of 3 nonnegative cubes in more than 1 way.

Original entry on oeis.org

216, 251, 344, 729, 855, 1009, 1072, 1366, 1457, 1459, 1520, 1674, 1728, 1729, 1730, 1737, 1756, 1763, 1793, 1854, 1945, 2008, 2072, 2241, 2414, 2456, 2458, 2729, 2736, 2752, 3060, 3391, 3402, 3457, 3500, 3592, 3599, 3655, 3744, 3745

Offset: 1

N. J. A. Sloane

nonn

G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 165.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Decomposition of first 10000 terms into cube triples

Cf. A001235, A003998, A008917, A025396, A025456, A025447.

Select[Range[4000], Length[PowersRepresentations[#, 3, 3]] > 1 &] (* Harvey P. Dale, Feb 03 2011 *)

is(n)=my(t); for(a=0, sqrtnint(n, 3), my(a3=a^3, c); for(b=0, min(a, sqrtnint(n-a3, 3)), if(ispower(n-a3-b^3, 3, &c) && c <= b && t++>1, return(1)))); 0 \\ Charles R Greathouse IV, Jul 02 2017

cp's OEIS Frontend

A001235 Taxi-cab numbers: sums of 2 cubes in more than 1 way.

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A024796 Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k.

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A025456 Number of partitions of n into 3 positive cubes.

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A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

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A025396 Numbers that are the sum of 3 positive cubes in exactly 2 ways.

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A309762 Numbers that are the sum of 3 nonzero 4th powers in more than one way.

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A024974 Numbers that are the sum of 3 distinct positive cubes in 2 or more ways.

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A025397 Numbers that are the sum of 3 positive cubes in exactly 3 ways.

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