A025407 -id:A025407 - cp's OEIS frontend

A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

5104, 9729, 12104, 12221, 12384, 13896, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687

Offset: 1

David W. Wilson

nonn

			a(1) = A230477(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3. - _Jonathan Sondow_, Oct 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A008917, A018787, A025331, A025397, A025402, A025407, A230477, A343968, A344239.

Select[ Range[ 50000], 2 < Length @ Cases[ PowersRepresentations[#, 3, 3], {?Positive, ?Positive, A008917%20by%20_Jonathan%20Sondow">?Positive}] &] (* adapted from Alcover's program for A008917 by _Jonathan Sondow, Oct 24 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

A008917(n) < a(n) <= A025397(n). - Jonathan Sondow, Oct 24 2013

{n: A025456(n) >= 3}. - R. J. Mathar, Jun 15 2018

A025457 Number of partitions of n into 4 positive cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

The first term > 1 is a(219) = 2. - Michel Marcus, Apr 23 2019

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Cf. A003108, A025455, A025456, A025403-A025407, A003327, A025420 (greedy inverse).

N:= 100;
A:= Array(0..N);
for a from 1 to floor(N^(1/3)) do
  for b from a to floor((N-a^3)^(1/3)) do
     for c from b to floor((N-a^3-b^3)^(1/3)) do
        for d from c to floor((N-a^3-b^3-c^3)^(1/3)) do
          n:= a^3 + b^3 + c^3 + d^3;
          A[n]:= A[n]+1;
od od od od:
seq(A[n],n=0..N); # Robert Israel, Aug 18 2014
A025457 := proc(n)
    local a,x,y,z,ucu ;
    a := 0 ;
    for x from 1 do
        if 4*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+3*y^3 > n then
                break;
            end if;
            for z from y do
                if x^3+y^3+2*z^3 > n then
                    break;
                end if;
                ucu := n-x^3-y^3-z^3 ;
                if isA000578(ucu) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

r[n_] := Reduce[0 < a <= b <= c <= d && n == a^3+b^3+c^3+d^3, {a, b, c, d}, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Table[a[n], {n, 0, 107}] (* Jean-François Alcover, Feb 26 2019 *)

a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Second offset from Michel Marcus, Apr 23 2019

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])

A343971 Numbers that are the sum of four positive cubes in four or more ways.

Original entry on oeis.org

1979, 2737, 3663, 4384, 4445, 4474, 4949, 5105, 5131, 5257, 5320, 5473, 5499, 5553, 5616, 5733, 5768, 5833, 5852, 5859, 6064, 6104, 6328, 6372, 6435, 6587, 6643, 6832, 6883, 6912, 6974, 7000, 7030, 7120, 7217, 7371, 7560, 7686, 7777, 7840, 8099, 8108, 8281, 8316, 8344, 8379, 8414, 8505, 8568, 8927, 9016, 9018

Offset: 1

David Consiglio, Jr., May 05 2021

nonn

			3663 = 1^3 + 10^3 + 11^3 + 11^3
     = 2^3 +  4^3 +  6^3 + 15^3
     = 2^3 +  9^3 +  9^3 + 13^3
     = 4^3 +  7^3 +  8^3 + 14^3
so 3663 is a term.

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

Cf. A025369, A025407, A343968, A343972, A343987, A344034, A344352.

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)]
for pos in cwr(power_terms,4):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 4])
for x in range(len(rets)):
    print(rets[x])

A344241 Numbers that are the sum of four fourth powers in three or more ways.

Original entry on oeis.org

16578, 43234, 49329, 53218, 54978, 57154, 93393, 106354, 107649, 108754, 138258, 151219, 160434, 168963, 173539, 177699, 178738, 181138, 183603, 185298, 195378, 195859, 196418, 197154, 197778, 201683, 202419, 209763, 211249, 216594, 217138, 223074, 234274, 235554, 235569, 236674, 237249, 237699

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			49329 = 2^4 + 2^4 + 12^4 + 13^4
      = 4^4 + 8^4 +  9^4 + 14^4
      = 6^4 + 9^4 + 12^4 + 12^4
so 49329 is a term of this sequence.

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

Cf. A025407, A309763, A344239, A344242, A344243, A344352, A345337.

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 >= 3])
for x in range(len(rets)):
    print(rets[x])

A025368 Numbers that are the sum of 4 nonzero squares in 3 or more ways.

Original entry on oeis.org

28, 42, 52, 55, 58, 60, 63, 66, 67, 70, 73, 75, 76, 78, 79, 82, 84, 85, 87, 90, 91, 92, 93, 95, 97, 98, 99, 100, 102, 103, 105, 106, 108, 109, 110, 111, 112, 114, 115, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 129, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141

Offset: 1

David W. Wilson

nonn

Index entries for sequences related to sums of squares

Cf. A025331, A025359, A025367, A025369, A025407, A344796.

{n: A025428(n) >=3}. Union of A025369 and A025359.- 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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A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

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A025457 Number of partitions of n into 4 positive cubes.

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

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

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A343971 Numbers that are the sum of four positive cubes in four or more ways.

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A344241 Numbers that are the sum of four fourth powers in three or more ways.

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A025368 Numbers that are the sum of 4 nonzero squares in 3 or more ways.

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

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