A003338 -id:A003338 - cp's OEIS frontend

A004814 Numbers that are the sum of 3 positive 11th powers.

3, 2050, 4097, 6144, 177149, 179196, 181243, 354295, 356342, 531441, 4194306, 4196353, 4198400, 4371452, 4373499, 4548598, 8388609, 8390656, 8565755, 12582912, 48828127, 48830174, 48832221, 49005273, 49007320, 49182419, 53022430

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)
204800049005272 is in the sequence as 204800049005272 = 3^11 + 5^11 + 20^11.
2518268235958260 is in the sequence as 2518268235958260 = 16^11 + 19^11 + 25^11.
3786934745885995 is in the sequence as 3786934745885995 = 10^11 + 19^11 + 26^11. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), 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), 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), A047700 (5, 2).

Removed incorrect program. - David A. Corneth, Aug 01 2020

A186653 Total number of positive integers below 10^n requiring 4 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 7, 48, 346, 3066, 27754, 260724, 2516312, 24744689, 245221669, 2442288495

Offset: 1

Martin Renner, Feb 25 2011

nonn

A114322(n) + A186649(n) + A186651(n) + a(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003338, A186654.

a(5)-a(9) from Lars Blomberg, May 08 2011

a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186654 Total number of n-digit numbers requiring 4 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 41, 298, 2720, 24688, 232970, 2255588, 22228377, 220476980, 2197066826

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + a(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003338, A186653.

a(n) = A186653(n) - A186653(n-1).

a(5)-a(11) from Giovanni Resta, Apr 26 2016

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 *)

A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

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, 16578, 16643, 16818, 17187, 17234, 17299, 17314, 17858, 18963, 19699

Offset: 1

Ilya Gutkovskiy, Aug 15 2019

nonn

			259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A003338, A018786, A025367, A025406, A309762, A344193, A344238, A344241, A344644.

N:= 10^5: # for terms <= N
V:= Vector(N, datatype=integer[4]):
for a from 1 while a^4 <= N do
  for b from 1 to a while a^4+b^4 <= N do
    for c from 1 to b while a^4 + b^4+ c^4 <= N do
      for d from 1 to c do
         v:= a^4+b^4+c^4+d^4;
         if v > N then break fi;
         V[v]:= V[v]+1
od od od od:
select(i -> V[i]>1, [$1..N]); # Robert Israel, Oct 07 2019

Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]

A344189 Numbers that are the sum of four fourth powers in exactly one way.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218, 1252, 1267, 1282, 1299, 1314, 1329, 1332, 1344, 1347, 1379, 1393

Offset: 1

David Consiglio, Jr., May 11 2021

nonn

Differs from A003338 at term 14 because 259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4

			34 is a member of this sequence because 34 = 1^4 + 1^4 + 2^4 + 2^4

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

Cf. A003338, A025403, A344188, A344190, A344193, A344642.

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 == 1])
for x in range(len(rets)):
    print(rets[x])

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

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)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), 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), 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), A047700 (5, 2).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

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

A176197 Sum of 4 distinct nonzero fourth powers.

Original entry on oeis.org

354, 723, 898, 963, 978, 1394, 1569, 1634, 1649, 1938, 2003, 2018, 2178, 2193, 2258, 2499, 2674, 2739, 2754, 3043, 3108, 3123, 3283, 3298, 3363, 3714, 3779, 3794, 3954, 3969, 4034, 4194, 4323, 4338, 4369, 4403, 4434, 4449, 4578, 4738, 4803, 4818, 4978

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 11 2010

nonn

1^4+2^4+3^4+4^4=354, 1^4+2^4+3^4+5^4=723, .., 2^4+3^4+4^4+5^4=978,..

Part of "Euler's Equation of degree four"

Subsequence of A003338.

# returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=aA176197 := proc(n)
    local a,i,j,k,l,res ;
    a := 0 ;
    for i from 1 do
        if i^4 > n then
            break ;
        end if;
        for j from i+1 do
            if i^4+j^4 > n then
                break ;
            end if;
            for k from j+1 do
                if i^4+j^4+k^4> n then
                    break;
                end if;
                res := n-i^4-j^4-k^4 ;
                if issqr(res) then
                    res := sqrt(res) ;
                    if issqr(res) then
                        l := sqrt(res) ;
                        if l > k then
                            a := a+1 ;
                        end if;
                    end if;
                end if;
            end do:
        end do:
    end do:
    a ;
end proc:
for n from 1 do
    if A176197(n) > 0 then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 17 2023

lst={};Do[Do[Do[Do[AppendTo[lst,a^4+b^4+c^4+d^4],{d,c+1,11}],{c,b+1,10}],{b,a+1,9}],{a,1,8}];Sort@lst

A047715 Numbers that are the sum of 4 but no fewer nonzero fourth powers.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

First differs from A003338 at term 64: A003338(64) = 1393 is also a term of A003337, so not a term here. - Michael S. Branicky, Apr 19 2021

Cf. A000583, A002377, A003338 (sum of 4), A003337 (sum of 3), A003336 (sum of 2), A344188, A344187.

limit = 1153
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n,)} if n in qds else set()
  return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
A003338s = set(n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1)
A003337s = set(n for n in range(3, limit+1) if len(findsums(n, 3)) >= 1)
A003336s = set(n for n in range(2, limit+1) if len(findsums(n, 2)) >= 1)
print(sorted(A003338s - A003337s - A003336s - qds)) # Michael S. Branicky, Apr 19 2021

Equals A003338 - A344188 - A344187 - A000583, where "-" denotes "set difference". - Sean A. Irvine, May 15 2021

cp's OEIS Frontend

A004814 Numbers that are the sum of 3 positive 11th powers.

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A186653 Total number of positive integers below 10^n requiring 4 positive biquadrates in their representation as sum of biquadrates.

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A186654 Total number of n-digit numbers requiring 4 positive biquadrates in their representation as sum of biquadrates.

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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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Mathematica

A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

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Maple

Mathematica

A344189 Numbers that are the sum of four fourth powers in exactly one way.

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Python

A003386 Numbers that are the sum of 8 nonzero 8th powers.

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Mathematica

Extensions

A176197 Sum of 4 distinct nonzero fourth powers.

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Maple

Mathematica

A047715 Numbers that are the sum of 4 but no fewer nonzero fourth powers.

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