A025404 -id:A025404 - cp's OEIS frontend

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

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

A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710

Offset: 1

Eric W. Weisstein

nonn

It appears that this sequence has 15416 terms, the last of which is 2243453. - Donovan Johnson, Jan 11 2013

From a(1) = 157 we see that c(n) = (number of ways n is the sum of 5 cubes) coincides with A010057 = characteristic function of cubes, up to n = 156. This sequence lists the numbers n for which c(n) = 2. See A003328 for c(n) > 0 and A048926 for c(n) = 1. - M. F. Hasler, Jan 04 2023

Donovan Johnson, Table of n, a(n) for n = 1..15416 (terms < 10^8)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003328 (sums of 5 positive cubes), A025404, A048926 (sum of 5 positive cubes in exactly 1 way), A048930, A294736, A343702, A343705, A344237.

Select[ Range[ 1000], (test = Length[ Select[ PowersRepresentations[#, 5, 3], And @@ (Positive /@ #)& ] ] == 2; If[test, Print[#]]; test)& ](* Jean-François Alcover, Nov 09 2012 *)

(waycount(n,numcubes,imax)={if(numcubes==0, !n, sum(i=1,imax, waycount(n-i^3,numcubes-1,i)))}); isA048927(n)=(waycount(n,5,floor(n^(1/3)))==2); \\ Michael B. Porter, Sep 27 2009

def ways (n, left = 5, last = 1):
  a = last; a3 = a**3; c = 0
  while a3 <= n-left+1:
    if left > 1:
       c += ways(n-a3, left-1, a)
    elif a3 == n:
       c += 1
    a += 1; a3 = a**3
  return c
for n in range (1,1000): # to print this sequence
  if ways(n)==2: print(n,end=", ") # in Python2 use, e.g.: print n,
# Minor edits by M. F. Hasler, Jan 04 2023

More terms from Walter Hofmann (walterh(AT)gmx.de), Jun 01 2000

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

A344193 Numbers that are the sum of four fourth powers in exactly two ways.

Original entry on oeis.org

259, 2674, 2689, 2754, 2929, 3298, 3969, 4144, 4209, 5074, 6579, 6594, 6659, 6769, 6834, 7203, 7874, 8194, 8979, 9154, 9234, 10113, 10674, 11298, 12673, 12913, 13139, 14674, 14689, 14754, 16563, 16643, 16818, 17187, 17234, 17299, 17314, 17858, 18963, 19699, 20658, 20739, 20979, 21154, 21219, 21329, 21363

Offset: 1

David Consiglio, Jr., May 11 2021

nonn

Differs from A309763 at term 32 because 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4

			2689 is a member of this sequence because 2689 = 2^4 + 2^4 + 4^4 + 7^4 = 2^4 + 3^4 + 6^4 + 6^4

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

Cf. A025404, A309763, A344189, A344192, A344237, A344242, A344645.

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,4):
    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])

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

A025405 Numbers that are the sum of 4 positive cubes in exactly 3 ways.

Original entry on oeis.org

1225, 1521, 1582, 1584, 1738, 1764, 2009, 2249, 2366, 2415, 2422, 2457, 2459, 2485, 2520, 2539, 2753, 2763, 2790, 2799, 3008, 3094, 3185, 3187, 3213, 3248, 3276, 3392, 3456, 3458, 3465, 3572, 3582, 3600, 3607, 3626, 3656, 3717, 3736, 3753, 3815, 3941

Offset: 1

David W. Wilson

nonn

Cf. A025359, A025397, A025404, A025407, A343705, A343972, A344242.

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

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

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A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

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

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A344193 Numbers that are the sum of four fourth powers in exactly two ways.

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

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A025405 Numbers that are the sum of 4 positive cubes in exactly 3 ways.

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