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

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

2,1

Comments

Numbers can be used more than once.

Links

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, International Journal of Number Theory 11 (8) (2015), 2505-2511.

Formula

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]

Mathematica

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 *)

Crossrefs

Cf. A191090, A191091.

Sequence in context: A172117 A217529 A284711 * A044210 A044591 A050255

Adjacent sequences: A193015 A193016 A193017 * A193019 A193020 A193021

Keyword

nonn,easy

Author

Remmert Borst, Jul 14 2011

Status

approved