A090461 Numbers k for which there exists a permutation of the numbers 1 to k such that the sum of adjacent numbers is a square.

15, 16, 17, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Author

T. D. Noe, Dec 01 2003

Keywords

• nonn

Comments

Conjecture: sequence includes all integers k > 24. See A090460 for the number of essentially different solutions.

It is now known that 25..299 are in the sequence, see the Numberphile 2 link. - Jud McCranie, Jan 11 2018

Every 25 <= k <= 2^20 is in the sequence and (71*25^m-1)/2 is also in the sequence for every m, hence this sequence is infinite, see Mersenneforum link for the proof; we give Hamiltonian cycle for these k values if k >= 32. - Robert Gerbicz, Jan 17 2017

The conjecture has been proved: every k >= 25 is in the sequence, moreover for k >= 32 there is a Hamiltonian cycle; see Mersenneforum topic for a code and deterministic algorithm to find a sequence. - Robert Gerbicz, Jan 21 2018

Examples

			See A071983.
		

Links

• Brady Haran and Matt Parker, The Square-Sum Problem, Numberphile video (2018)
• Brady Haran, Matt Parker, and Charlie Turner, The Square-Sum Problem (extra footage) - Numberphile 2 (2018)
• HexagonVideos, Numberphile's Square-Sum Problem was solved!, YouTube video, 2023.
• Mersenneforum, The Square-Sum problem

Crossrefs

Cf. A071983, A071984 (number of circular solutions), A090460.

Cf. A078107 (k for which there is no solution).

Programs

Maple

F:= proc(n)
uses GraphTheory;
local edg, G;
edg:= select(t -> issqr(t[1]+t[2]),{seq(seq({i,j},i=1..j-1),j=1..n)}) union {seq({i,n+1},i=1..n)};
G:= Graph(n+1,edg);
IsHamiltonian(G)
end proc:
select(F, [$1..50]); # Robert Israel, Jun 05 2015

Mathematica

Join[{15, 16, 17, 23}, Range[25, 100]] (* Paolo Xausa, May 28 2024 *)

Extensions

a(31)-a(69) from Donovan Johnson, Sep 14 2010