A025396 -id:A025396 - 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

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

A025395 Numbers that are the sum of 3 positive cubes in exactly 1 way.

Original entry on oeis.org

3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, 80, 81, 92, 99, 118, 127, 129, 134, 136, 141, 153, 155, 160, 179, 190, 192, 197, 216, 218, 225, 232, 244, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434, 440

Offset: 1

David W. Wilson

A025456(a(n)) = 1. - Reinhard Zumkeller, Apr 23 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubic Number.
Index entries for sequences related to sums of cubes

Cf. A003072, A024981, A025321, A025396, A025403, A338667, A344188.

Reap[For[n = 1, n <= 500, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 1, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

A025404 Numbers that are the sum of 4 positive cubes in exactly 2 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, 1251, 1352, 1367, 1368, 1374, 1375, 1393, 1397, 1423, 1430, 1458, 1460, 1465, 1467, 1484, 1486

Offset: 1

David W. Wilson

nonn

Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A025358, A025396, A025403, A025405, A025406, A048927, A344193.

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

A344192 Numbers that are the sum of three fourth powers in exactly two ways.

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, 364658, 384833, 439922, 457488

Offset: 1

David Consiglio, Jr., May 11 2021

nonn

Differs from A309762 at term 59 because 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4

			16562 is a member of this sequence because 16562 = 1^4 + 9^4 + 10^4 = 5^4 + 6^4 + 11^4

David Consiglio, Jr., Table of n, a(n) for n = 1..10000

Cf. A025396, A309762, A344188, A344193, A344240.

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 == 2])
for x in range(len(rets)):
    print(rets[x])

A343708 Numbers that are the sum of two positive cubes in exactly two ways.

Original entry on oeis.org

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, 805688, 842751, 885248, 886464

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A001235 at term 455 because 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3 = A011541(3). Thus, this term is not in this sequence but is in A001235.

			13832 is in this sequence because 13832 = 2^3 + 24^3 = 18^3 + 20^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A001235, A025285, A025396, A338667, A344804.

Select[Range@70000,Length@Select[PowersRepresentations[#,2,3],FreeQ[#,0]&]==2&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

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

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

A219329 Numbers that can be expressed as the sum of three nonnegative cubes in three ways.

Original entry on oeis.org

5104, 5832, 9288, 9729, 10261, 10773, 12104, 12221, 12384, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24416, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 39528, 40060, 40097, 40832, 40851, 41033, 41040, 41364

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 11 2013

nonn

Index of A051343 = 9, superset of index of A025456 = 3.

Subset of A001239.

			a(1) = 5104 = 1^3+12^3+15^3 = 2^3+10^3+16^3 = 9^3+10^3+15^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Decomposition of the first 10000 terms into 3 sets of cube triples
Sequences related to sums of squares and cubes

Other sums of cubes: A025402, A025398, A024974, A001239, A008917.
Cf. A025456, A051343.
Cf. A025396.

Select[Range[42000],Length[PowersRepresentations[#,3,3]]==3&] (* Harvey P. Dale, Sep 28 2016 *)

cp's OEIS Frontend

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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A025395 Numbers that are the sum of 3 positive cubes in exactly 1 way.

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A025404 Numbers that are the sum of 4 positive cubes in exactly 2 ways.

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A344192 Numbers that are the sum of three fourth powers in exactly two ways.

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A343708 Numbers that are the sum of two positive cubes in exactly two ways.

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

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