A100719 Size of the largest subset of {1,2,...,n} such that no two distinct elements differ by a perfect square.

1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20

Offset

1,3

Comments

Prompted by a question about the rate of growth of this sequence asked by Sujith Vijay (sujith(AT)EDEN.RUTGERS.EDU) to the Number Theory List.

The problem can be thought of as finding a maximum independent set in a graph with node set {1, 2, ..., n} and an edge (i, j) if |i - j| is a square. - Rob Pratt.

The index of the first occurrence of m is A210570(m). - Glen Whitney, 2015 Aug 30

References

Bloom, Thomas F., and James Maynard. "A new upper bound for sets with no square differences." Compositio Mathematica 158.8 (2022): 1777-1798; also arXiv:2011.13266, Feb 24 2021.

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. Analyse Math. 31 (1977), 204-256.

A. Sárközy, On difference sets of sequences of integers II, Annales Univ. Sci. Budapest, Sectio Math.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..410 (terms n = 1..100 from Rob Pratt)

A. Balog, J. Pelikan, J. Pintz and E. Szemeredi, Difference sets without kappa-th powers, Acta Math. Hungar. 65 (1994), no. 2, 165-187.

Fausto A. C. Cariboni, Sets of maximal span that yield a(n) for n = 3..314, Nov 28 2018.

Ben Green and Mehtaab Sawhney, Improved bounds for the Furstenberg-Sárközy theorem, arXiv preprint arXiv:2411.17448 [math.NT], 2024.

J. Pintz, W. L. Steiger and E. Szemeredi, On Sets of Natural Numbers Whose Difference Set Contains No Squares, J. London. Math. Soc. 37, 1988, 219-231.

I. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205-209.

A. Sárközy, On difference sets of sequences of integers I, Acta Mathematica Academiae Scientiarum Hungarica, March 1978, Volume 31, Issue 1, pp 125-149.

A. Sárközy, On difference sets of sequences of integers III, Acta Mathematica Academiae Scientiarum Hungarica, September 1978, Volume 31, Issue 3, pp 355-386.

Formula

a(n) >> n^0.733 (I. Ruzsa, Period. Math. Hungar. 15 (1984), no. 3, 205-209). The best upper bound appears to be O(N (log n)^(- c log log log log N)) due to Pintz, Steiger and Szemeredi (J. London. Math. Soc. 37, 1988, 219-231). - Sujith Vijay, Sep 18 2007

A. Sárközy worked on this problem. There is also the following result of A. Balog, J. Pelikan, J. Pintz, E. Szemeredi: the size of largest squarefree difference sets = O(N / (log N)^(log log log log N / 4)). - Tsz Ho Chan (tchan(AT)MEMPHIS.EDU), Sep 19 2007

Green & Sawhney improve the upper bound to a(n) << n exp(-(log n)^c) for any c < 1/4. - Charles R Greathouse IV, Nov 28 2024

Crossrefs

Cf. A210570.

Cf. A131752, A131753, A131754.

Sequence in context: A355028 A061375 A029920 * A057354 A172476 A172267

Adjacent sequences: A100716 A100717 A100718 * A100720 A100721 A100722

Keyword

nonn

Author

N. J. A. Sloane, Sep 17 2007

Extensions

Computed via integer programming by Rob Pratt, Sep 17 2007

Status

approved