A191091 -id:A191091 - cp's OEIS frontend

A191090 a(n)^2 is the largest square required when writing n as a partition of squares.

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, 2, 2, 4, 2, 3, 2, 6, 2, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 4, 7, 5, 2, 4, 2, 3, 2, 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

Offset: 1

T. D. Noe, Jul 18 2011

nonn

If we write n as the sum of nondecreasing squares, then a(n) is the largest value such that n = a(n)^2 + ....

			a(8) = 2 because 8 = 2^2 + 2^2.

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

Cf. A191091, A193018.

Table[Last[Union[First /@ Union /@ (DeleteCases[#, 0] & /@ PowersRepresentations[n, 11, 2])]], {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]

cp's OEIS Frontend

A191090 a(n)^2 is the largest square required when writing n as a partition of squares.

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