A025395 -id:A025395 - cp's OEIS frontend

A025456 Number of partitions of n into 3 positive cubes.

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

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

A344188 Numbers that are the sum of three fourth powers in exactly one way.

Original entry on oeis.org

3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177, 2403, 2418, 2433, 2483, 2498, 2546, 2563, 2593, 2608, 2658

Offset: 1

David Consiglio, Jr., May 11 2021

nonn

Differs from A003337 and A047714 at term 60 because 2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, see A309762.

			33 is a member of this sequence because 33 = 1^4 + 2^4 + 2^4

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

Cf. A003337, A025395, A344187, A344189, A344192, A344641.

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 == 1])
for x in range(len(rets)):
    print(rets[x])

A025399 Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.

Original entry on oeis.org

36, 73, 92, 99, 134, 153, 160, 190, 197, 216, 225, 244, 251, 281, 288, 307, 342, 349, 352, 368, 371, 378, 405, 408, 415, 434, 469, 476, 495, 521, 532, 540, 547, 560, 567, 577, 584, 586, 603, 623, 638, 645, 664, 684, 701, 729, 736, 738, 755, 757, 764, 792, 794, 801, 820

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A025395 (not necessarily distinct), A024973.

Reap[For[n = 1, n <= 1000, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 && Length[#] == Length[Union[#]] &]; If[pr != {} && Length[pr] == 1, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

A024975 MINUS A024974. - R. J. Mathar, May 28 2008

{n: A025469(n) = 1}. - R. J. Mathar, Jun 15 2018

A025403 Numbers that are the sum of 4 positive cubes in exactly 1 way.

Original entry on oeis.org

4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 81, 82, 88, 89, 93, 100, 107, 108, 119, 126, 128, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 217, 224, 226, 233, 240, 243, 245, 254, 256, 261, 266, 271, 280

Offset: 1

David W. Wilson

nonn

Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003327, A025357, A025395, A025399, A025404, A048926, A344189.

Reap[For[n = 1, n <= 300, n++, pr = Select[ PowersRepresentations[n, 4, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 1, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

{n: A025457(n) = 1}. - R. J. Mathar, Jun 15 2018

A338667 Numbers that are the sum of two positive cubes in exactly one way.

Original entry on oeis.org

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, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1736

Offset: 1

David Consiglio, Jr., Apr 22 2021

This sequence differs from A003325 at term 61: A003325(61) = 1729 is the famous Ramanujan taxicab number and is excluded from this sequence because it is the sum of two cubes in two distinct ways.

			35 is a term of this sequence because 2^3 + 3^3 = 8 + 27 = 35 and this is the one and only way to express 35 as the sum of two cubes.

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

Cf. A003325, A025284, A025395, A343708, A344187.

Select[Range@2000,Length[s=PowersRepresentations[#,2,3]]==1&&And@@(#>0&@@@s)&] (* Giorgos Kalogeropoulos, Apr 24 2021 *)

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
from bisect import bisect_left as bisect
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,1000)]
for pos in cwr(power_terms,2):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 1])
for x in range(len(rets)):
    print(rets[x])

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A025456 Number of partitions of n into 3 positive cubes.

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

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A344188 Numbers that are the sum of three fourth powers in exactly one way.

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A025399 Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.

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A025403 Numbers that are the sum of 4 positive cubes in exactly 1 way.

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A338667 Numbers that are the sum of two positive cubes in exactly one way.

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