A003386 -id:A003386 - cp's OEIS frontend

A003072 Numbers that are the sum of 3 positive cubes.

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, 251, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434

Offset: 1

N. J. A. Sloane, David W. Wilson

A119977 is a subsequence; if m is a term then there exists at least one k>0 such that m-k^3 is a term of A003325. - Reinhard Zumkeller, Jun 03 2006
A025456(a(n)) > 0. - Reinhard Zumkeller, Apr 23 2009
Davenport proved that a(n) << n^(54/47 + e) for every e > 0. - Charles R Greathouse IV, Mar 26 2012

			a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.

K. D. Bajpai, Table of n, a(n) for n = 1..12955 (first 1000 terms from T. D. Noe)
H. Davenport, Sums of three positive cubes, J. London Math. Soc., 25 (1950), 339-343. Coll. Works III p. 999.
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Subsequence of A004825.
Cf. A003325, A024981, A057904 (complement), A010057, A000578, A023042 (subsequence of cubes).
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
- squares: A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2);
- cubes: A003325 (2, 3), A003072 (3, 3), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3);
- fourth powers: A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4);
- fifth powers: A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5);
- sixth powers: A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6);
- seventh powers: A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7);
- eighth powers: A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003386 (8, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8);
- ninth powers: A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9);
- tenth powers: A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10);
- eleventh powers: A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11).

a003072 n = a003072_list !! (n-1)
a003072_list = filter c3 [1..] where
   c3 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (> 0) $ map (x -) $ a003325_list
-- Reinhard Zumkeller, Mar 24 2012

isA003072 := proc(n)
    local x,y,z;
    for x from 1 do
        if 3*x^3 > n then
            return false;
        end if;
        for y from x do
            if x^3+2*y^3 > n then
                break;
            end if;
            if isA000578(n-x^3-y^3) then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 1000 do
    if isA003072(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 23 2016

Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jean-François Alcover, Apr 29 2011 *)
With[{upto=500},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2,3]]]^3,3]],#<=upto&]] (* Harvey P. Dale, Oct 25 2021 *)

sum(n=1,11,x^(n^3),O(x^1400))^3 /* Then [i|i<-[1..#%],polcoef(%,i)] gives the list of powers with nonzero coefficient. - M. F. Hasler, Aug 02 2020 */

list(lim)=my(v=List(),k,t); lim\=1; for(x=1,sqrtnint(lim-2,3), for(y=1, min(sqrtnint(lim-x^3-1,3),x), k=x^3+y^3; for(z=1,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

{n: A025456(n) >0}. - R. J. Mathar, Jun 15 2018

Incorrect program removed by David A. Corneth, Aug 01 2020

A003390 Sum of 12 nonzero 8th powers.

Original entry on oeis.org

12, 267, 522, 777, 1032, 1287, 1542, 1797, 2052, 2307, 2562, 2817, 3072, 6572, 6827, 7082, 7337, 7592, 7847, 8102, 8357, 8612, 8867, 9122, 9377, 13132, 13387, 13642, 13897, 14152, 14407, 14662, 14917, 15172, 15427, 15682, 19692, 19947, 20202, 20457

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1890948 is in the sequence as 1890948 = 2^8 + 2^8 + 2^8 + 4^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 5^8 + 5^8 + 5^8.
2338951 is in the sequence as 2338951 = 1^8 + 1^8 + 1^8 + 1^8 + 1^8 + 3^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 6^8.
3841896 is in the sequence as 3841896 = 1^8 + 1^8 + 1^8 + 2^8 + 3^8 + 3^8 + 3^8 + 3^8 + 4^8 + 5^8 + 6^8 + 6^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (replacing a b-file that missed terms)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2), A003072 (3, 3), A003325 (2, 3), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003386 (8, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11).

A003390_upto(N=1e5, n=12, p=8)={my(P=[x^p|x<-[1..sqrtnint(N-n+1, p)]], S=P); while(n--, S=Set(concat([[x+y|y<-S, x+y<=N]|x<-P]))); S} \\ M. F. Hasler, Jul 03 2025

Removed incorrect program, offset corrected by David A. Corneth, Aug 01 2020

A336725 A(n,k) is the n-th number that is a sum of k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 10, 8, 4, 5, 19, 17, 10, 5, 6, 36, 34, 24, 13, 6, 7, 69, 67, 49, 29, 17, 7, 8, 134, 132, 98, 64, 36, 18, 8, 9, 263, 261, 195, 129, 84, 43, 20, 9, 10, 520, 518, 388, 258, 160, 99, 55, 25, 10, 11, 1033, 1031, 773, 515, 321, 247, 114, 62, 26, 11, 12, 2058, 2056, 1542, 1028, 642, 384, 278, 129, 66, 29, 12

Offset: 1

Alois P. Heinz, Aug 01 2020

			Square array A(n,k) begins:
   1,  2,  3,   4,   5,   6,    7,    8,    9,   10, ...
   2,  5, 10,  19,  36,  69,  134,  263,  520, 1033, ...
   3,  8, 17,  34,  67, 132,  261,  518, 1031, 2056, ...
   4, 10, 24,  49,  98, 195,  388,  773, 1542, 3079, ...
   5, 13, 29,  64, 129, 258,  515, 1028, 2053, 4102, ...
   6, 17, 36,  84, 160, 321,  642, 1283, 2564, 5125, ...
   7, 18, 43,  99, 247, 384,  769, 1538, 3075, 6148, ...
   8, 20, 55, 114, 278, 734,  896, 1793, 3586, 7171, ...
   9, 25, 62, 129, 309, 797, 2193, 2048, 4097, 8194, ...
  10, 26, 66, 164, 340, 860, 2320, 6568, 4608, 9217, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-11 give: A000027, A000404, A003072, A003338, A003350, A003362, A003374, A003386, A003398, A004810, A004822.
Rows n=1-3 give: A000027, A052944, A145071.
Main diagonal gives A000337.
Cf. A336820.

A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end,
      proc(n, k) option remember; local b; b:=
        proc(x, y) option remember; `if`(x=0, {0}, `if`(y<1, {},
          {b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]}))
        end;
        while nops(w(k)) < n do forget(b);
          l(k):= [l(k)[], (nops(l(k))+1)^k];
          w(k):= sort([select(h-> h

nmax = 12;
pow[n_, k_] := IntegerPartitions[n, {k}, Range[n^(1/k) // Ceiling]^k];
col[k_] := col[k] = Reap[Module[{j = k, n = 1, p}, While[n <= nmax, p = pow[j, k]; If[p =!= {}, Sow[j]; n++]; j++]]][[2, 1]];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2020 *)

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A003072 Numbers that are the sum of 3 positive cubes.

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A003390 Sum of 12 nonzero 8th powers.

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A336725 A(n,k) is the n-th number that is a sum of k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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