A194769 -id:A194769 - 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

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

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

cp's OEIS Frontend

A001422 Numbers which are not the sum of distinct squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A003999 Sums of distinct nonzero 4th powers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A194768 Sum of distinct positive fifth powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions