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
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, arXiv:1408.1435 [math.NT], 2014-2015; International Journal of Number Theory 11 (8) (2015), 2505-2511.
Programs
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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 *)
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]
Comments