A343702 -id:A343702 - cp's OEIS frontend

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

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

A343704 Numbers that are the sum of five positive cubes in three or more ways.

Original entry on oeis.org

766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044, 1070, 1105, 1108, 1142, 1145, 1161, 1168, 1226, 1233, 1252, 1259, 1289, 1350, 1368, 1376, 1424, 1431, 1439, 1441, 1457, 1461, 1487, 1492, 1494, 1522, 1529, 1531, 1538, 1548, 1550, 1555, 1568, 1583, 1585, 1587, 1590, 1592, 1593, 1594, 1609, 1611, 1613, 1639

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A343705 at term 20 because 1252 = 1^3+1^3+5^3+5^3+10^3= 1^3+2^3+3^3+6^3+10^3 = 3^3+3^3+7^3+7^3+8^3 = 3^3+4^3+6^3+6^3+9^3. Thus this term is in this sequence but not A343705.

			827 is a member of this sequence because 827 = 1^3 + 4^3 + 5^3 + 5^3 + 8^3 = 2^3 + 2^3 + 5^3 + 7^3 + 7^3 = 2^3 + 3^3 + 4^3 + 6^3 + 8^3.

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

Cf. A025407, A343702, A343705, A344034, A344243, A344796, A345512.

Select[Range@2000,Length@Select[PowersRepresentations[#,5,3],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(1,50)]#n
for pos in cwr(power_terms,5):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 3])#s
for x in range(len(rets)):
    print(rets[x])

A344238 Numbers that are the sum of five fourth powers in two or more ways.

Original entry on oeis.org

260, 275, 340, 515, 884, 1555, 2595, 2660, 2675, 2690, 2705, 2755, 2770, 2835, 2930, 2945, 3010, 3185, 3299, 3314, 3379, 3554, 3923, 3970, 3985, 4050, 4115, 4145, 4160, 4210, 4225, 4290, 4355, 4400, 4465, 4594, 4769, 4834, 5075, 5090, 5155, 5265, 5330, 5395, 5440, 5505, 5570, 5699, 6370, 6545, 6580, 6595, 6610

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			340 = 1^4 + 1^4 + 1^4 + 3^4 + 4^4
    = 2^4 + 3^4 + 3^4 + 3^4 + 3^4
so 340 is a term of this sequence.

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

Cf. A003339, A309763, A342685, A343702, A344237, A344243, A345559.

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, 5):
    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])

A345511 Numbers that are the sum of six cubes in two or more ways.

Original entry on oeis.org

158, 165, 184, 221, 228, 235, 247, 254, 256, 261, 268, 273, 275, 280, 282, 284, 287, 291, 294, 306, 310, 313, 317, 324, 331, 332, 343, 345, 347, 350, 352, 362, 369, 371, 373, 376, 378, 380, 385, 387, 388, 392, 395, 399, 404, 406, 408, 411, 418, 425, 430, 432

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A003329, A048930, A343702, A344806, A345512, A345520, A345559.

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)]
for pos in cwr(power_terms, 6):
    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])

A344795 Numbers that are the sum of five squares in two or more ways.

Original entry on oeis.org

20, 29, 32, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Sean A. Irvine, May 28 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..20000

Cf. A025367, A047700, A294736, A343702, A344796, A344806.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A343704 Numbers that are the sum of five positive cubes in three or more ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A344238 Numbers that are the sum of five fourth powers in two or more ways.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A345511 Numbers that are the sum of six cubes in two or more ways.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A344795 Numbers that are the sum of five squares in two or more ways.

Views

Author

Keywords

Links

Crossrefs