A025398 -id:A025398 - cp's OEIS frontend

A025331 Numbers that are the sum of 3 nonzero squares in 3 or more ways.

54, 66, 81, 86, 89, 99, 101, 110, 114, 126, 129, 131, 134, 146, 149, 150, 153, 161, 162, 166, 171, 173, 174, 179, 182, 185, 186, 189, 194, 198, 201, 206, 209, 216, 219, 221, 222, 225, 227, 230, 233, 234, 237, 241, 242, 243, 245, 246, 249, 251, 254, 257, 258, 261, 264

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..7639
Index entries for sequences related to sums of squares

Cf. A024796, A025294, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025368, A025398, A025427.

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

A008917 Numbers that are the sum of 3 positive cubes in more than one way.

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

Offset: 1

N. J. A. Sloane and J. H. Conway

Of course reordering the terms does not count.

A025456(a(n)) > 1. [Reinhard Zumkeller, Apr 23 2009]

			a(2) = 1009 = 1^3+2^3+10^3 = 4^3+6^3+9^3.

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Christian N. K. Anderson, Decomposition of first 10000 terms into multiple cube triples.

Cf. A001235, A003072, A024796, A025396, A025398, A025406, A309762.

Select[Range[4450], 1 < Length @ Cases[PowersRepresentations[#, 3, 3], {?Positive, ?Positive, ?Positive}] &]  (* _Jean-François Alcover, Apr 04 2011 *)

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)>1, print1(n, ", ")); n++) \\ Derek Orr, Aug 27 2015

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

A343968 Numbers that are the sum of three positive cubes in four or more ways.

Original entry on oeis.org

13896, 40041, 44946, 52200, 53136, 58995, 76168, 82278, 93339, 94184, 105552, 110683, 111168, 112384, 112832, 113400, 143424, 149416, 149904, 161568, 167616, 169560, 171296, 175104, 196776, 197569, 208144, 216126, 221696, 222984, 224505, 235808, 240813, 252062, 255312, 262683, 262781, 266031

Offset: 1

David Consiglio, Jr., May 05 2021

nonn

			44946 =  7^3 + 12^3 + 35^3
      =  9^3 + 17^3 + 34^3
      = 11^3 + 24^3 + 31^3
      = 16^3 + 17^3 + 33^3
so 44946 is a term.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A023051, A025332, A025398, A343967, A343969, A343971, A344277.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,50)]
for pos in cwr(power_terms,3):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 4])
for x in range(len(rets)):
    print(rets[x])

A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

Original entry on oeis.org

1225, 1521, 1582, 1584, 1738, 1764, 1979, 2009, 2249, 2366, 2415, 2422, 2457, 2459, 2485, 2520, 2539, 2737, 2753, 2763, 2790, 2799, 3008, 3094, 3185, 3187, 3213, 3248, 3276, 3392, 3456, 3458, 3465, 3572, 3582, 3600, 3607, 3626, 3656, 3663, 3717, 3736

Offset: 1

David W. Wilson

nonn

Cf. A025368, A025398, A025405, A025406, A343704, A343971, A344241.

{n: A025457(n) >= 3}. - R. J. Mathar, Jun 15 2018

A018787 Numbers that are the sum of two positive cubes in at least three ways (all solutions).

Original entry on oeis.org

87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, 3080802816, 3235261176, 3499524728, 3623721192, 3877315533, 4750893000, 5544709352, 5602516416

Offset: 1

David W. Wilson, Aug 15 1996

nonn

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
R. K. Guy, Unsolved Problems in Number Theory, D1.

Ray Chandler, Table of n, a(n) for n=1..100000
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]

Cf. A001235, A003825, A023051, A025294, A025398, A344804.

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 *)

A230477 Smallest number that is the sum of n positive n-th powers in >= n ways.

Original entry on oeis.org

1, 50, 5104, 236674, 9006349824, 82188309244

Offset: 1

Jonathan Sondow, Oct 22 2013

Does a(6) exist? For which values of n does a(n) exist? Is there a proof that a(n) < a(n+1) when both exist?

			1 = 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
236674 = 1^4 + 2^4 + 7^4 + 22^4 = 3^4 + 6^4 + 18^4 + 19^4 = 7^4 + 14^4 + 16^4 + 19^4 = 8^4 + 16^4 + 17^4 + 17^4.
9006349824 = 8^5 + 34^5 + 62^5 + 68^5 + 92^5 = 8^5 + 41^5 + 47^5 + 79^5 + 89^5 = 12^5 + 18^5 + 72^5 + 78^5 + 84^5 = 21^5 + 34^5 + 43^5 + 74^5 + 92^5 = 24^5 + 42^5 + 48^5 + 54^5 + 96^5.
82188309244 = 1^6 + 9^6 + 29^6 + 44^6 + 55^6 + 60^6 = 2^6 + 12^6 + 25^6 + 51^6 + 53^6 + 59^6 = 5^6 + 23^6 + 27^6 + 44^6 + 51^6 + 62^6 = 10^6 + 16^6 + 41^6 + 45^6 + 51^6 + 61^6 = 12^6 + 23^6 + 33^6 + 34^6 + 55^6 + 61^6 = 15^6 + 23^6 + 31^6 + 36^6 + 53^6 + 62^6.

A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, NY, 1966, pp. 162-165, 290-291.
R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

Open Directory Project, Equal Sums of Like Powers

a(2) = A048610(2), a(3) = A025398(1), a(4) = A219921(1).

Cf. A146756 (smallest number that is the sum of n distinct positive n-th powers in exactly n ways), A230561 (smallest number that is the sum of two positive n-th powers in >= n ways), A091414 (smallest number that is the sum of n positive n-th powers in >= 2 ways).

a(n) <= A146756(n), with equality at least for n = 1, 3, 5 and inequality at least for n = 2, 4.

a(n) >= A091414(n) for n > 1, with equality at least for n = 2 and inequality at least for n = 3, 4, 5.

a(5) from Donovan Johnson, Oct 23 2013

a(6) from Michael S. Branicky, May 09 2021

A344239 Numbers that are the sum of three fourth powers in three or more ways.

Original entry on oeis.org

811538, 1733522, 2798978, 3750578, 4614722, 5978882, 6573938, 7303842, 8878898, 12771458, 12984608, 13760258, 14677362, 15601698, 15916082, 16196193, 17868242, 20621042, 21556178, 22349522, 22673378, 25190802, 25589858, 27736352, 29969282, 30623138, 33998258, 39765362, 41532498, 44048498

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			2798978 =  6^4 + 31^4 + 37^4
        =  9^4 + 29^4 + 38^4
        = 13^4 + 26^4 + 39^4
so 2798978 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..14213

Cf. A025398, A309762, A344240, A344241, A344277.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1,50)]
for pos in cwr(power_terms,3):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 3])
for x in range(len(rets)):
    print(rets[x])

A025402 Numbers that are the sum of 3 distinct positive cubes in 3 or more ways.

Original entry on oeis.org

5104, 9729, 12104, 12384, 13896, 14175, 17604, 17928, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 42056, 43408, 44946, 45144, 46593, 46684

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..11601

Cf. A025398.

{n: A025469(n) >= 3}. - R. J. Mathar, Jun 15 2018

cp's OEIS Frontend

A025331 Numbers that are the sum of 3 nonzero squares in 3 or more ways.

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A025456 Number of partitions of n into 3 positive cubes.

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A008917 Numbers that are the sum of 3 positive cubes in more than one way.

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

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A343968 Numbers that are the sum of three positive cubes in four or more ways.

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A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

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A018787 Numbers that are the sum of two positive cubes in at least three ways (all solutions).

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A230477 Smallest number that is the sum of n positive n-th powers in >= n ways.

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A344239 Numbers that are the sum of three fourth powers in three or more ways.

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