A004845 -id:A004845 - cp's OEIS frontend

A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 3, 5, 5, 0, 1, 2, 3, 8, 8, 6, 0, 1, 2, 3, 4, 9, 9, 7, 0, 1, 2, 3, 4, 16, 10, 10, 8, 0, 1, 2, 3, 4, 5, 17, 16, 13, 9, 0, 1, 2, 3, 4, 5, 32, 18, 17, 16, 10, 0, 1, 2, 3, 4, 5, 6, 33, 19, 24, 17, 11, 0, 1, 2, 3, 4, 5, 6, 64, 34, 32, 27, 18, 12

Offset: 1

Alois P. Heinz, Aug 04 2020

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,   0,   0,   0,  0,  0, ...
   1,  1,  1,  1,  1,  1,   1,   1,   1,  1,  1, ...
   2,  2,  2,  2,  2,  2,   2,   2,   2,  2,  2, ...
   3,  4,  3,  3,  3,  3,   3,   3,   3,  3,  3, ...
   4,  5,  8,  4,  4,  4,   4,   4,   4,  4,  4, ...
   5,  8,  9, 16,  5,  5,   5,   5,   5,  5,  5, ...
   6,  9, 10, 17, 32,  6,   6,   6,   6,  6,  6, ...
   7, 10, 16, 18, 33, 64,   7,   7,   7,  7,  7, ...
   8, 13, 17, 19, 34, 65, 128,   8,   8,  8,  8, ...
   9, 16, 24, 32, 35, 66, 129, 256,   9,  9,  9, ...
  10, 17, 27, 33, 36, 67, 130, 257, 512, 10, 10, ...

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

Columns k=1-11 give: A001477(n-1), A001481, A004825, A004833, A004845, A004857, A004869, A004881, A004893, A004905, A004917.
A(n+j,n) for j=0-3 give: A001477(n-1), A000027, A000079, A000051.
Cf. A336725.

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 or y<1, {},
          {0, 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

b[n_, k_, i_, t_] := b[n, k, i, t] = n == 0 || i > 0 && t > 0 && (b[n, k, i - 1, t] || i^k <= n && b[n - i^k, k, i, t - 1]);
A[n_, k_] := A[n, k] = Module[{m}, For[m = 1 + If[n == 1, -1, A[n - 1, k]], !b[m, k, m^(1/k) // Floor, k], m++]; m];
Table[A[n, 1+d-n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, Dec 03 2020, using Alois P. Heinz's code for columns *)

A(n,k) = n-1 for n <= k+1.

A342688 Numbers that are the sum of five positive fifth powers in exactly three ways.

Original entry on oeis.org

13124675, 28055699, 50043937, 52679923, 53069024, 55097976, 57936559, 60484744, 62260463, 62445305, 70211956, 73133026, 79401728, 80368962, 84766210, 88512249, 93288865, 98824300, 106993391, 113055482, 117173891, 120968132, 123383875, 126416258, 131106051, 131529588, 132022925

Offset: 1

David Consiglio, Jr., May 18 2021

nonn

Differs from A342687:
287618651 =  8^5 + 21^5 + 27^5 + 27^5 + 48^5
          =  9^5 + 13^5 + 26^5 + 37^5 + 46^5
          = 11^5 + 12^5 + 23^5 + 41^5 + 44^5
          = 11^5 + 20^5 + 22^5 + 30^5 + 48^5.
So 287618651 is a term of A342687 but not a term of this sequence.
[Corrected by Patrick De Geest, Dec 28 2024]

			50043937 =  6^5 + 16^5 + 18^5 + 24^5 + 33^5
         =  7^5 + 13^5 + 21^5 + 23^5 + 33^5
         = 11^5 + 13^5 + 13^5 + 29^5 + 31^5
so 50043937 is a term of this sequence.
[Corrected by _Patrick De Geest_, Dec 28 2024]

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A003350, A004845, A342685, A342686, A342687, A344244, A344519, A344518, A345863, A345864, A346257, A346358.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**5 for x in range(1, 500)]
for pos in cwr(power_terms, 5):
    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])

A344519 Numbers that are the sum of five positive fifth powers in exactly four ways.

Original entry on oeis.org

287618651, 1386406515, 1763135232, 2494769760, 2619898293, 3096064443, 3291315732, 3749564512, 4045994624, 5142310350, 5183605813, 5658934676, 5880926107, 7205217018, 7401155424, 7691215599, 8429499101, 8926086432, 9051501568, 9203796832, 9254212901

Offset: 1

David Consiglio, Jr., May 21 2021

nonn

Differs from A344518 at term 20 because
9006349824 =  8^5 + 34^5 + 62^5 + 68^5 + 92^5
           =  8^5 + 41^5 + 47^5 + 79^5 + 89^5
           = 12^5 + 18^5 + 72^5 + 78^5 + 84^5
           = 21^5 + 34^5 + 43^5 + 74^5 + 92^5
           = 24^5 + 42^5 + 48^5 + 54^5 + 96^5.

			287618651 is a term because
287618651 =  8^5 + 21^5 + 27^5 + 27^5 + 48^5
          =  9^5 + 13^5 + 26^5 + 37^5 + 46^5
          = 11^5 + 12^5 + 23^5 + 41^5 + 44^5
          = 11^5 + 20^5 + 22^5 + 30^5 + 48^5.
[Corrected by _Patrick De Geest_, Dec 28 2024]

Sean A. Irvine, Table of n, a(n) for n = 1..4857

Cf. A003350, A004845, A342685, A342686, A342687, A342688, A344244, A344355, A344518, A345863, A345864, A346257, A346358, A346359.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**5 for x in range(1, 500)]
for pos in cwr(power_terms, 5):
    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])

A004842 Numbers that are the sum of at most 2 positive 5th powers.

Original entry on oeis.org

0, 1, 2, 32, 33, 64, 243, 244, 275, 486, 1024, 1025, 1056, 1267, 2048, 3125, 3126, 3157, 3368, 4149, 6250, 7776, 7777, 7808, 8019, 8800, 10901, 15552, 16807, 16808, 16839, 17050, 17831, 19932, 24583, 32768, 32769, 32800, 33011, 33614, 33792, 35893, 40544

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 159 terms from Vincenzo Librandi)

Cf. A004845.

Select[Table[n, {n, 0, 50000}], Length[PowersRepresentations[#, 2, 5]] > 0 &] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

T=thueinit('z^5+1);is(n)=n==0 || #select(v->min(v[1],v[2])>=0, thue(T, n))>0 \\ Charles R Greathouse IV, Nov 30 2014

is(n)=for(m=sqrtnint(n\2,5), sqrtnint(n,5), if(ispower(n-m^5,5), return(1))); 0 \\ Charles R Greathouse IV, Nov 30 2014

list(lim)=my(v=List(),n5); for(n=0,sqrtnint(lim\=1,5), n5=n^5; for(m=0, min(sqrtnint(lim-n5,5),n), listput(v, n5+m^5))); Set(v) \\ Charles R Greathouse IV, Nov 30 2014

a(n) << n^(5/2). - Charles R Greathouse IV, Nov 30 2014

A004843 Numbers that are the sum of at most 3 positive 5th powers.

Original entry on oeis.org

0, 1, 2, 3, 32, 33, 34, 64, 65, 96, 243, 244, 245, 275, 276, 307, 486, 487, 518, 729, 1024, 1025, 1026, 1056, 1057, 1088, 1267, 1268, 1299, 1510, 2048, 2049, 2080, 2291, 3072, 3125, 3126, 3127, 3157, 3158, 3189, 3368, 3369, 3400, 3611, 4149, 4150, 4181, 4392

Offset: 1

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 200 terms from Vincenzo Librandi)

Cf. A004842, A004845.

b:= proc(n, i, t) option remember; n=0 or i>0 and t>0
      and (b(n, i-1, t) or i^5<=n and b(n-i^5, i, t-1))
    end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, -1, a(n-1))
      while not b(k, iroot(k, 5), 3) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 16 2016

Select[Table[n, {n, 0, 6000}], Length[PowersRepresentations[#, 3, 5]] > 0 &] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

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A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A342688 Numbers that are the sum of five positive fifth powers in exactly three ways.

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A344519 Numbers that are the sum of five positive fifth powers in exactly four ways.

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A004842 Numbers that are the sum of at most 2 positive 5th powers.

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A004843 Numbers that are the sum of at most 3 positive 5th powers.

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