A191090 -id:A191090 - cp's OEIS frontend

A191091 Least number of squares required when writing n as a sum of squares with the smallest square as large as possible.

1, 2, 3, 1, 2, 3, 4, 2, 1, 2, 3, 3, 2, 3, 4, 1, 3, 2, 3, 2, 4, 3, 4, 3, 1, 4, 3, 4, 2, 5, 4, 2, 3, 2, 5, 1, 4, 3, 6, 2, 2, 4, 3, 3, 2, 5, 4, 3, 1, 2, 5, 2, 2, 3, 6, 3, 3, 2, 3, 4, 2, 3, 4, 1, 2, 3, 3, 3, 3, 3, 4, 2, 4, 2, 3, 4, 3, 3, 4, 2, 1, 4, 3, 4, 2, 3, 4

Offset: 1

T. D. Noe, Jul 21 2011

nonn

The smallest value in the sum is A191090. All sums have at most 6 terms for n <= 1000. Is this the largest possible value? For n up to 1000, only 39, 55, 143, 543, and 564 need 6 terms.

			The sum for 30 gives the first sum having 5 terms: 4 + 4 + 4 + 9 + 9.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A191090, A193018.

Table[s=DeleteCases[#, 0]& /@ PowersRepresentations[n,11,2]; t=Last[Union[First /@ Union /@ s]]; Min @@ Length /@ Select[s, #[[1]] == t &], {n, 100}]

A193018 The largest integer that cannot be written as the sum of squares of integers larger than n.

Original entry on oeis.org

23, 87, 119, 201, 312, 376, 455, 616, 760, 840, 1055, 1136, 1248, 1472, 1719, 1959, 2064, 2472, 2764, 2976, 3264, 3407, 3584, 4032, 4336, 4848, 4992, 5088, 5523, 5900, 6112, 6624, 7360, 7680, 7680, 8448, 8960, 9152, 9856, 10208, 11136, 11904, 12256, 12256

Offset: 2

Remmert Borst, Jul 14 2011

Numbers can be used more than once.

Giovanni Resta, Table of n, a(n) for n = 2..100
Ken Dutch and Christy Rickett, Conductors for sets of large integer squares, Notes on Number Theory and Discrete Mathematics Vol. 18 (2012), No. 1, 16-21.
Alessio Moscariello, On integers which are representable as sums of large squares, arXiv:1408.1435 [math.NT], 2014-2015; International Journal of Number Theory 11 (8) (2015), 2505-2511.

Cf. A191090, A191091.

a[n_] := Block[{k = 4, f}, While[ (n+k)^2 <= (f = FrobeniusNumber[ Range[ n, n+k]^2]), k++]; f]; a /@ Range[2, 45] (* Giovanni Resta, Jun 13 2016 *)

a(n) < n^4 + 6n^3 + 11n^2 + 6n by Sylvester's theorem. [Charles R Greathouse IV, Jul 14 2011]
a(n) = o(n^{2+e}) for all e > 0, according to Dutch and Rickett. [Jeffrey Shallit, Mar 17 2021]
a(n) = O(n^2), according to Moscariello.  [Jeffrey Shallit, Mar 17 2021]

A241898 a(n) is the largest integer such that n = a(n)^2 + ... is a decomposition of n into a sum of at most four nondecreasing squares.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 4, 2, 3, 1, 2, 2, 2, 1, 2, 5, 2, 3, 2, 2, 1, 2, 4, 2, 3, 1, 6, 2, 2, 1, 2, 4, 2, 3, 2, 3, 1, 2, 4, 7, 5, 1, 4, 2, 3, 1, 2, 4, 3, 3, 2, 5, 2, 3, 8, 4, 4, 3, 4, 2, 3, 2, 6, 4, 5, 5, 3, 4, 2, 3, 4, 9, 4, 3, 4, 6, 5, 2

Offset: 1

Moshe Shmuel Newman, May 15 2014

This differs from A191090 only for n>=30 because 30 cannot be written as a sum of at most four squares without using 1^2, but 30 can be written as a sum of five nondecreasing squares: 2^2 + 2^2 + 2^2 + 3^2 + 3^2, making A191090(30)=2.

By Lagrange's Theorem every number can be written as a sum of four squares. Can the same be said of the set of {a^2|a is any integer not equal to 7}? From the data that I have, it would seem that a(n) is greater than 7 for all n>599. If this could be proved, it would only remain to check if all the numbers up to 599 can be written as the sum of 4 squares none of which is 7^2.

			30 can be written as the sum of at most 4 nondecreasing squares in the following ways: 1^2 + 2^2 + 5^2 or 1^2 + 2^2 + 3^2 + 4^2. Therefore, a(30)=1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A191090.

b:= proc(n, i, t) option remember; n=0 or t>0 and
       i^2<=n and (b(n, i+1, t) or b(n-i^2, i, t-1))
    end:
a:= proc(n) local k;
      for k from isqrt(n) by -1 do
        if b(n, k, 4) then return k fi
      od
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 25 2014

For[i=0,i<=7^4,i++,a[i]={}];
For[i1=0,i1<=7,i1++,
For[i2=0,i2<=7,i2++,
For[i3=0,i3<=7,i3++,
For[i4=0,i4<=7,i4++,
sumOfSquares=i1^2+i2^2+i3^2+i4^2;
smallestSquare=Min[DeleteCases[{i1,i2,i3,i4},0]];
a[sumOfSquares]=Union[{smallestSquare},a[sumOfSquares]] ]]]];
Table[Max[a[i]],{i,1,50}]

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A191091 Least number of squares required when writing n as a sum of squares with the smallest square as large as possible.

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A193018 The largest integer that cannot be written as the sum of squares of integers larger than n.

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A241898 a(n) is the largest integer such that n = a(n)^2 + ... is a decomposition of n into a sum of at most four nondecreasing squares.

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