A001661 -id:A001661 - cp's OEIS frontend

A001422 Numbers which are not the sum of distinct squares.

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128

Offset: 1

N. J. A. Sloane, Jeff Adams (jeff.adams(AT)byu.net)

This is the complete list (Sprague).

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 222.

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
T. Sillke, Not the sum of distinct squares
R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51, (1948), 289-290.
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Complement of A003995. Subsequence of A004441.
Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A053614, A121405 (corresponding sequences for triangular and pentagonal numbers)
Cf. A033461, A276517.
Cf. A001476, A046039, A194768, A194769 for 3rd, 4th, 5th, 6th powers.
Cf. A001661.

nn=50; t=Rest[CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

select( is_A001422(n,m=n)={m^2>n&& m=sqrtint(n); n!=m^2&&!while(m>1,isSumOfSquares(n-m^2,m--)&&return)}, [1..128]) \\ M. F. Hasler, Apr 21 2020

A030052 Smallest number whose n-th power is a sum of distinct smaller positive n-th powers.

Original entry on oeis.org

3, 5, 6, 15, 12, 25, 40, 84, 47, 63, 68, 81, 102, 95, 104, 162, 123

Offset: 1

Richard C. Schroeppel

Sprague has shown that for any n, all sufficiently large integers are the sum of distinct n-th powers. Sequence A001661 lists the largest number not of this form, so we know that a(n) is less than or equal to the next larger n-th power. - M. F. Hasler, May 25 2020

a(18) <= 200, a(19) <= 234, a(20) <= 242; for more upper bounds see the Al Zimmermann's Programming Contests link: The "Final Report" gives exact solutions for n = 16 through 30; those for n = 16 and 17 have been confirmed to be minimal by Jeremy Sawicki. - M. F. Hasler, Jul 20 2020

			3^1 = 2^1 + 1^1, and there is no smaller solution given that the r.h.s. is the smallest possible sum of distinct positive powers.
For n = 2, one sees immediately that 3 is not a solution (3^2 > 2^2 + 1^2) and one can check that 4^2 isn't equal to Sum_{x in A} x^2 for any subset A of {1, 2, 3}. Therefore, the well known hypotenuse number 5 (cf. A009003) with 5^2 = 4^2 + 3^2 provides the smallest possible solution.
a(17) = 123 since 123^17 = Sum {3, 5, 7, 8, 9, 11, 13, 16, 17, 19, 30, 33, 34, 35, 38, 40, 41, 43, 51, 52, 54, 55, 58, 59, 60, 63, 66, 69, 70, 71, 72, 73, 75, 76, 81, 86, 87, 88, 89, 90, 92, 95, 98, 106, 107, 108, 120}^17, with obvious notation. [Solution found by Jeremy Sawicki on July 3, 2020, see Al Zimmermann's Programming Contests link.] - _M. F. Hasler_, Jul 18 2020
For more examples, see the link.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Various authors, How each power is decomposed
Eric Weisstein's World of Mathematics, Diophantine Equation.
Al Zimmermann, Sum Of Powers: Final Report, Al Zimmermann's Programming Contests, April-July 2020

Cf. A078607, A332065, A332097, A332101.

Other sequences defined by sums of distinct n-th powers: A001661, A364637.

A030052(n, m=n\/log(2)+1, s=0)={if(!s, until(A030052(n, m, (m+=1)^n),), s < 2^n || s > (m+n+1)*m^n\(n+1), m=s<2, m=min(sqrtnint(s, n), m); s==m^n || until( A030052(n, m-1, s-m^n) || (s>=(m+n)*(m-=1)^n\(n+1) && !m=0), )); m} \\ Does exhaustive search to find the least solution m. Use optional 2nd arg to specify a starting value for m. Calls itself with nonzero 3rd (optional) argument: in this case, returns nonzero iff s is the sum of powers <= m^n. - For illustration only: takes very long already for n = 8 and n >= 10. - M. F. Hasler, May 25 2020

a(n) <= A001661(n)^(1/n) + 1. - M. F. Hasler, May 25 2020

a(n) >= A332101(n) = A078607(n)+2 (conjectured). - M. F. Hasler, May 25 2020

a(8)-a(10) found by David W. Wilson
a(11) from Al Zimmermann, Apr 07 2004
a(12) from Al Zimmermann, Apr 13 2004
a(13) from Manol Iliev, Jan 04 2010
a(14) and a(15) from Manol Iliev, Apr 28 2011
a(16) and a(17) due to Jeremy Sawicki, added by M. F. Hasler, Jul 20 2020

A332065 Infinite square array where row n lists the integers whose n-th power is the sum of distinct n-th powers of positive integers; read by falling antidiagonals.

Original entry on oeis.org

3, 4, 5, 5, 7, 6, 6, 9, 9, 15, 7, 10, 12, 25, 12, 8, 11, 13, 27, 23, 25, 9, 12, 14, 29, 24, 28, 40, 10, 13, 15, 30, 28, 32, 43, 84, 11, 14, 16, 31, 29, 34, 44, 85, 47, 12, 15, 17, 33, 30, 35, 45, 86, 49, 63, 13, 16, 18, 35, 31, 36, 46, 87, 52, 64, 68

Offset: 1

M. F. Hasler, Mar 31 2020

Each row contains all sufficiently large integers (Sprague). Sequences A001422, A001476, A046039, A194768, A194769, ... mention the largest number which can't be written as sum of distinct n-th powers for n = 2, 3, 4, 5, 6, ...; see also A001661. Sequence A332066 gives the number of positive integers not in row n.

All positive multiples of any T(n,k) appear later in that row (because if s^n = Sum_{x in S} x^n, then (k*s)^n = Sum_{x in k*S} x^n).

			The table reads: (Entries from where on T(n,k+1) = T(n,k)+1 are marked by *.)
   n | k=1    2    3    4    5    6    7    8    9   10   11   12   13  ...
  ---+---------------------------------------------------------------------
   1 |   3*   4    5    6    7    8    9   10   11   12   13   14   15  ...
   2 |   5    7    9*  10   11   12   13   14   15   16   17   18   19  ...
   3 |   6    9   12*  13   14   15   16   17   18   19   20   21   22  ...
   4 |  15   25   27   29   30   31   33   35   37   39   41   43   45* ...
   5 |  12   23   24   28*  29   30   31   32   33   34   35   36   37  ...
   6 |  25   28   32   34*  35   36   37   38   39   40   41   42   43  ...
   7 |  40   43*  44   45   46   47   48   49   50   51   52   53   54  ...
   8 |  84*  85   86   87   88   89   90   91   92   93   94   95   96  ...
   9 |  47   49   52*  53   54   55   56   57   58   59   60   61   62  ...
  10 |  63*  64   65   66   67   68   69   70   71   72   73   74   75  ...
  11 |  68   73*  74   75   76   77   78   79   80   81   82   83   84  ...
  ...| ...
Row 1: 3^1 = 2^1 + 1^1, 4^1 = 3^1 + 1^1, 5^1 = 4^1 + 1^1, 6^1 = 5^1 + 1^1, ...
Row 2: 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, 9^2 = 8^2 + 4^2 + 1^2, ...
Row 3: 6^3 = 5^3 + 4^3 + 3^3, 9^3 = 8^3 + 6^3 + 1, 12^3 = 10^3 + 8^3 + 6^3, ...
Row 4: 15^4 = Sum {14, 9, 8, 6, 4}^4, 25^4 = Sum {21, 20, 12, 10, 8, 6, 2}^4, ...
See the link for other rows.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Various authors, Decomposition of T(n,1)^n = A030052(n)^n.

Cf. A030052 (first column), A001661.
Cf. A009003 (hypotenuse numbers; subsequence of row 2).
Cf. A001422, A001476, A046039, A194768, A194769.
Cf. A332066.

M332065=Map(); A332065(n,m,r)={if(r, if( m<2^n||m>r^n*(r+n+1)\(n+1), m<2, r=min(sqrtnint(m,n),r), m==r^n || while( !A332065(n,m-r^n,r-=1) && (mA004736(n),n=A002260(n)]; mapisdefined(M332065,[n,m],&r), r, n<2, m+2, r=if(m>1,A332065(n,m-1),n+2); until(A332065(n, (r+=1)^n, r-1),); mapput(M332065,[n,m],r); r)} \\ Calls itself with nonzero (optional) 3rd argument to find by exhaustive search whether r can be written as sum of distinct powers <= m^n. (Comment added by M. F. Hasler, May 25 2020)

T(1,k) = 2 + k for all k. (Indeed, s^1 = (s-1)^1 + 1 and s-1 > 1 for s > 2.)
T(2,k) = 6 + k for all k >= 3. (Use s^2 = (s-1)^2 + 2*s-1 and A001422, A009003.)
T(3,k) = 9 + k for all k >= 3. (Use max A001476 = 12758 < 24^3.)
T(4,k) = 32 + k for all k >= 13. (Use max A046039 < 48^4.)
T(5,k) = 24 + k for all k >= 4. (Use max(N \ A194768) < 37^5.)
T(6,k) = 30 + k for all k >= 4. (Use max(N \ A194769) < 48^6.)
T(7,k) = 41 + k for all k >= 2.
T(9,k) = 49 + k for all k >= 3.

More terms from M. F. Hasler, Jul 19 2020

A276517 Indices k such that A276516(k) = 0.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 41, 43, 44, 45, 46, 47, 48, 53, 54, 60, 61, 67, 70, 72, 74, 76, 79, 82, 84, 87, 90, 92, 93, 96, 105, 106, 107, 108, 111, 112, 114, 117, 122, 128, 133, 135, 139, 141, 148, 159

Offset: 1

Vaclav Kotesovec, Dec 12 2016

nonn

This is different from A001422, first difference: a(14) = 25, A001422(14) = 27.

Conjecture: for k > 7169 there are no more terms in this sequence (tested for k < 10000000).

			3 is in the sequence because A276516(3) = 0
4 is not in the sequence because A276516(4) = -1
4222 is in the sequence because A276516(4222) = 0
7169 is in the sequence because A276516(7169) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..173

Cf. A001422, A001661, A276516, A279486, A279487.

nn = 100; A276516 = Rest[CoefficientList[Series[Product[(1-x^(k^2)), {k, nn}], {x, 0, nn^2}], x]]; Select[Range[nn^2], A276516[[#]]==0&]
nmax = 10000; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; A276516 = Take[poly, {2, nmax+1}]; Select[Range[nmax], A276516[[#]]==0&]

A121571 Largest number that is not the sum of n-th powers of distinct primes.

Original entry on oeis.org

6, 17163, 1866000

Offset: 1

T. D. Noe, Aug 08 2006

As stated by Sierpinski, H. E. Richert proved a(1) = 6. Dressler et al. prove a(2) = 17163.
Fuller & Nichols prove T. D. Noe's conjecture that a(3) = 1866000. They also prove that 483370 positive numbers cannot be written as the sum of cubes of distinct primes. - Robert Nichols, Sep 08 2017
Noe conjectures that a(4) = 340250525752 and that 31332338304 positive numbers cannot be written as the sum of fourth powers of distinct primes. - Charles R Greathouse IV, Nov 04 2017

			a(1) = 6 because only the numbers 1, 4 and 6 are not the sum of distinct primes.

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964, p. 143-144.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.
Robert E. Dressler, Louis Pigno and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, (2015), #15.10.5.
H. E. Richert, Über Zerfällungen in ungleiche Primzahlen, Math. Z. 52 no. 1 (1948), 342-343.

Cf. A231407 (numbers that are not the sum of distinct primes).
Cf. A121518 (numbers that are not the sum of squares of distinct primes).
Cf. A213519 (numbers that are the sum of cubes of distinct primes).
Cf. A001661 (integers instead of primes).

a(1) = A231407(3), a(2) = A121518(2438). - Jonathan Sondow, Nov 26 2013

A279486 Indices k such that A279484(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77

Offset: 1

Vaclav Kotesovec, Dec 13 2016

nonn

This is different from A001476, first difference: a(450) = 540, A001476(450) = 542.

Conjecture: for k > 353684 there are no more terms in this sequence (tested for k < 1000000).

			3 is in the sequence because A279484(3) = 0
8 is not in the sequence because A279484(8) = -1
344739 is in the sequence because A279484(344739) = 0
353684 is in the sequence because A279484(353684) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..5216

Cf. A001476, A001661, A276517, A279484.

nn = 10; A279484 = Rest[CoefficientList[Series[Product[(1-x^(k^3)), {k, nn}], {x, 0, nn^3}], x]]; Select[Range[nn^3], A279484[[#]]==0&]
nmax = 1000; nn = Floor[nmax^(1/3)]+1; poly = ConstantArray[0, nn^3 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^3 + 1]], {j, nn^3, k^3, -1}];, {k, 2, nn}]; A279484 = Take[poly, {2, nmax+1}]; Select[Range[nmax], A279484[[#]]==0&]

A003999 Sums of distinct nonzero 4th powers.

Original entry on oeis.org

1, 16, 17, 81, 82, 97, 98, 256, 257, 272, 273, 337, 338, 353, 354, 625, 626, 641, 642, 706, 707, 722, 723, 881, 882, 897, 898, 962, 963, 978, 979, 1296, 1297, 1312, 1313, 1377, 1378, 1393, 1394, 1552, 1553, 1568, 1569, 1633, 1634, 1649, 1650, 1921, 1922

Offset: 1

N. J. A. Sloane

5134240 is the largest positive integer not in this sequence. - Jud McCranie

If we tightened the sequence requirement so that the sum was of more than one 4th power, we would remove exactly 32 4th powers from the terms: row 4 of A332065 indicates which 4th powers would remain. After a(1) = 1, the next number in this sequence that is in the analogous sequences for cubes and squares is a(24) = 881 = A364637(4). - Peter Munn, Aug 01 2023

The Penguin Dictionary of Curious and Interesting Numbers, David Wells, entry 5134240.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A046039 (complement).
Cf. A003995, A003997, A194768, A194769 (analogs for squares, cubes, 5th and 6th powers).
Cf. A001661, A332065, A364637.
A217844 is a subsequence.

(1+x)*(1+x^16)*(1+x^81)*(1+x^256)*(1+x^625)*(1+x^1296)*(1+x^2401)*(1+x^4096)*(1+x^6561)*(1+x^10000)

max = 2000; f[x_] := Product[1 + x^(k^4), {k, 1, 10}]; Exponent[#, x]& /@ List @@ Normal[Series[f[x], {x, 0, max}]] // Rest (* Jean-François Alcover, Nov 09 2012, after Charles R Greathouse IV *)

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/4),1+x^(n^4),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}; \\ Charles R Greathouse IV, Sep 02 2011

For n > 4244664, a(n) = n + 889576. - Charles R Greathouse IV, Sep 02 2011

A194768 Sum of distinct positive fifth powers.

Original entry on oeis.org

1, 32, 33, 243, 244, 275, 276, 1024, 1025, 1056, 1057, 1267, 1268, 1299, 1300, 3125, 3126, 3157, 3158, 3368, 3369, 3400, 3401, 4149, 4150, 4181, 4182, 4392, 4393, 4424, 4425, 7776, 7777, 7808, 7809, 8019, 8020, 8051, 8052, 8800, 8801, 8832, 8833, 9043, 9044, 9075, 9076

Offset: 1

Charles R Greathouse IV, Sep 02 2011

nonn

From Peter Munn, Aug 02 2023: (Start)
67898771 = A001661(5) is the largest number not in the sequence.
After a(1) = 1, the next term that is in all the analogous sequences for smaller powers is a(35) = 7809 = A364637(5).
If we tightened the sequence requirement so that the sum was of more than one 5th power, we would remove exactly 24 5th powers from the terms: row 5 of A332065 indicates which 5th powers would remain.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000584 (5th powers), A001661, A332065, A364637.
Cf. A003997, A003999, A194769 (analogs for 3rd, 4th and 6th powers).
A217845 is a subsequence.

N:= 2*10^4: # to get all terms <= N
S:= {0}:
for i from 1 while i^5 <= N do
  S:= select(`<=`, map(`+`,S,i^5),N) union S
od:
sort(convert(S minus {0},list)); # Robert Israel, Jun 26 2019

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/5),1+x^(n^5),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}; \\ Charles R Greathouse IV, Sep 02 2011

For n > 53986089, a(n) = n + 13912682. [Charles R Greathouse IV, Sep 02 2011]

Name qualified by Peter Munn, Aug 02 2023

A194769 Sum of distinct nonzero sixth powers.

Original entry on oeis.org

1, 64, 65, 729, 730, 793, 794, 4096, 4097, 4160, 4161, 4825, 4826, 4889, 4890, 15625, 15626, 15689, 15690, 16354, 16355, 16418, 16419, 19721, 19722, 19785, 19786, 20450, 20451, 20514, 20515, 46656, 46657, 46720, 46721, 47385, 47386, 47449, 47450, 50752, 50753, 50816

Offset: 1

Charles R Greathouse IV, Sep 02 2011

See A001661 for a proof of the formula. - M. F. Hasler, May 15 2020
From Peter Munn, Aug 02 2023: (Start)
11146309947 = A001661(6) is the largest number not in the sequence.
After a(1) = 1, the next term that is in all the analogous sequences for smaller powers is a(86) = 134067 = A364637(6).
If we tightened the sequence requirement so that the sum was of more than one 6th power, we would remove exactly 30 6th powers from the terms: row 6 of A332065 indicates which 6th powers would remain.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001014, A001661, A332065, A364637.
A217846 is a subsequence.
Cf. A003997, A003999, A194768 (analogs for 3rd, 4th and 5th powers).

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/6),1+x^(n^6),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}

For n > 9108736851, a(n) = n + 2037573096.

More terms from David A. Corneth, Apr 21 2020

Name qualified by Peter Munn, Aug 02 2023

A279487 Indices k such that A279485(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Vaclav Kotesovec, Dec 13 2016

nonn

This is different from A046039, first difference: a(14328) = 14979, A046039(14328) = 14981.
Conjecture: Last term is a(1040799) = 64674419. For k > 64674419 there are no more terms in this sequence (tested for k < 150000000).
Last terms are: 30082710, 30345655, 30358709, 30530388, 30982210, 31463972, 32369456, 32374194, 32594966, 32658048, 32780596, 32875172, 32997892, 33135812, 33440935, 33647428, 34086978, 34112787, 34629875, 35535908, 35638453, 36081828, 36140868, 36945332, 39218566, 39581363, 40364547, 40491526, 41235157, 43853600, 47973011, 57353782, 57767766, 64674419

			3 is in the sequence because A279485(3) = 0
16 is not in the sequence because A279485(16) = -1
57767766 is in the sequence because A279485(57767766) = 0
64674419 is in the sequence because A279485(64674419) = 0

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000

Cf. A001661, A046039, A279485.

Cf. A276517, A279486.

nn = 10; A279485 = Rest[CoefficientList[Series[Product[(1-x^(k^4)), {k, nn}], {x, 0, nn^4}], x]]; Select[Range[nn^4], A279485[[#]]==0&]
nmax = 10000; nn = Floor[nmax^(1/4)]+1; poly = ConstantArray[0, nn^4 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^4 + 1]], {j, nn^4, k^4, -1}];, {k, 2, nn}]; A279485 = Take[poly, {2, nmax+1}]; Select[Range[nmax], A279485[[#]]==0&]

cp's OEIS Frontend

A001422 Numbers which are not the sum of distinct squares.

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A030052 Smallest number whose n-th power is a sum of distinct smaller positive n-th powers.

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A332065 Infinite square array where row n lists the integers whose n-th power is the sum of distinct n-th powers of positive integers; read by falling antidiagonals.

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A276517 Indices k such that A276516(k) = 0.

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A121571 Largest number that is not the sum of n-th powers of distinct primes.

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A279486 Indices k such that A279484(k) = 0.

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A003999 Sums of distinct nonzero 4th powers.

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A194768 Sum of distinct positive fifth powers.