A025469 -id:A025469 - cp's OEIS frontend

A024975 Sums of three distinct positive cubes.

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

Clark Kimberling

nonn

Subsequence of A003072. Equals A024973 if duplicates of repeated entries are removed. - R. J. Mathar, Apr 13 2008

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A122723 (primes in here), A025399-A025402, A025411 (4 distinct positive cubes).

Total/@Subsets[Range[10]^3,{3}]//Union (* Harvey P. Dale, Aug 22 2021 *)

list(lim)=my(v=List(),x3,t); lim\=1; for(x=3,sqrtnint(lim-9,3), x3=x^3; for(y=2,min(x-1,sqrtnint(lim-x3-1,3)), t=x3+y^3; for(z=1,min(y-1,sqrtnint(lim-t,3)), listput(v,t+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 20 2016

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

Verified by Don Reble, Nov 19 2006

A025419 Least sum of 3 distinct positive cubes in exactly n ways.

Original entry on oeis.org

36, 1009, 5104, 13896, 161568, 1296378, 2016496, 2562624, 14926248, 34012224, 69190848, 150547032, 119095488, 1204376256, 952763904, 1592865000, 3974344704, 2176782336, 10077696000, 2985984000, 36330467328, 30723115968, 23887872000, 17414258688, 72825163776, 75686967000, 204141384000, 62099136000, 139314069504, 245784927744, 80621568000, 191102976000, 272097792000, 373248000000

Offset: 1

David W. Wilson

nonn

a(36) = 496793088000. a(37) = 980922617856. a(38) = 209584584000. a(39) = 644972544000. a(42) = 970299000000. - Donovan Johnson, Nov 13 2010

Cf. A025418.

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

a(6)-a(34) from Donovan Johnson, Nov 13 2010

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

A025400 Numbers that are the sum of 3 distinct positive cubes in exactly 2 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, 5130, 5435, 5472, 5643, 5706, 5825, 5832, 6183, 6301, 6642, 6848, 6904

Offset: 1

David W. Wilson

nonn

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

Cf. A025397 (not necessarily distinct)

Reap[ For[n = 1, n <= 7000, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 && Length[#] == Length[Union[#]] &] ; If[pr != {} && Length[pr] == 2, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)
Module[{nn=20},Select[Tally[Total/@Subsets[Range[nn]^3,{3}]],#[[2]]==2 && #[[1]]<= nn^3-9&][[All,1]]]//Union (* Harvey P. Dale, Apr 26 2020 *)

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

A025468 a(n) is the number of partitions of n into 2 distinct positive cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

In other words, number of solutions to the equation n = x^3 + y^3 with x > y > 0. The first value > 1 is a(1729) = 2. - Antti Karttunen, Aug 29 2017

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to sums of cubes

Cf. A000578, A048766, A025455, A025465, A025469.

(define (A025468 n) (let loop ((x (A048766 n)) (s 0)) (let* ((x3 (A000578 x)) (y3 (- n x3))) (if (<= x3 y3) s (loop (- x 1) (+ s (if (and (> y3 0) (= (A000578 (A048766 y3)) y3)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

From Antti Karttunen, Aug 28-29 2017: (Start)
a(n) = A025465(n) - A025469(n).
a(n) <= A025455(n).
(End)

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

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

A025401 Numbers that are the sum of 3 distinct positive cubes in exactly 3 ways.

Original entry on oeis.org

5104, 9729, 12104, 12384, 14175, 17604, 17928, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 36288, 38259, 39339, 39376, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 42056, 43408, 45144, 46593, 46684, 46747, 46808, 47683

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..11181

Cf. A025397 (not necessarily distinct)

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

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

A025465 Number of partitions of n into 3 distinct nonnegative cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

In other words, number of solutions to the equation n = x^3 + y^3 + z^3 with x > y > z >= 0. - Antti Karttunen, Aug 29 2017

			From _Antti Karttunen_, Aug 29 2017: (Start)
For n = 9 there is one solution: 9 = 2^3 + 1^3 + 0^3, thus a(9) = 1.
For n = 855 there are two solutions: 855 = 9^3 + 5^3 + 1^3 = 8^3 + 7^3 + 0^3, thus a(855) = 2. This is also the first point where sequence attains value greater than one.
(End)
From _Harvey P. Dale_, Sep 30 2018: (Start)
In addition to 855, the following numbers attain the value of 2: 1009, 1072, 1366, 1457, and there are 73 more such numbers less than 10000.
The first two numbers to attain the value of 3 are 5104 and 9729.
There are no numbers up to 10000 that attain a value greater than 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..17073
Index entries for sequences related to sums of cubes

Cf. A025468, A025469, A001239.

Table[Length[FindInstance[{n==x^3+y^3+z^3,x>y>z>=0},{x,y,z},Integers,5]],{n,0,110}] (* Harvey P. Dale, Sep 30 2018 *)

A025465(n) = { my(s=0); for(x=0,n,if(ispower(x,3),for(y=x+1,n-x,if(ispower(y,3),for(z=y+1,n-(x+y),if((ispower(z,3)&&(x+y+z)==n),s++)))))); (s); }; \\ Antti Karttunen, Aug 29 2017

a(n) = A025468(n) + A025469(n).

A347362 Smallest number which can be decomposed into exactly n sums of three distinct positive cubes, but cannot be decomposed into more than one such sum containing the same cube.

Original entry on oeis.org

36, 1009, 12384, 82278, 746992, 5401404, 15685704, 26936064, 137763072, 251066304, 857520000, 618817536, 3032856000, 2050677000, 6100691904, 36013192704, 16405416000, 96569712000, 48805535232, 131243328000, 611996202000, 201153672000

Offset: 1

Gleb Ivanov, Aug 29 2021

No cube should appear in two or more sums. 5104 = 15^3 + 10^3 + 9^3 = 15^3 + 12^3 + 1^3 = 16^3 + 10^3 + 2^3 is not a(3), because 15^3 appears in more than one sum.

			a(1) = 36 = 1^3 + 2^3 + 3^3.
a(2) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.
a(3) = 12384 = 1^3 + 6^3 + 23^3 = 2^3 + 12^3 + 22^3 = 15^3 + 16^3 + 17^3.

Gleb Ivanov, Rust program for large terms.
Gleb Ivanov, Python program.

Cf. A025333, A025419, A025469, A025398.

Cf. A343968, A343967, A345088, A345087, A345119, A345152.

Monitor[Do[k=1;While[Length@Union@Flatten[p=PowersRepresentations[k,3,3]]!=n*3||Length@p!=n||MemberQ[Flatten@p,0],k++];Print@k,{n,10}],k] (* Giorgos Kalogeropoulos, Sep 03 2021 *)

a(13)-a(15) from Jon E. Schoenfield, Sep 02 2021

a(16)-a(22) from Gleb Ivanov, Sep 12 2021

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A024975 Sums of three distinct positive cubes.

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A025419 Least sum of 3 distinct positive cubes in exactly n ways.

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

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A025400 Numbers that are the sum of 3 distinct positive cubes in exactly 2 ways.

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A025468 a(n) is the number of partitions of n into 2 distinct positive cubes.

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

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

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A025401 Numbers that are the sum of 3 distinct positive cubes in exactly 3 ways.

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A025465 Number of partitions of n into 3 distinct nonnegative cubes.

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