A071984 Square loops: the number of circular permutations (reversals not counted as different) of the numbers 1 to n such that the sum of any two consecutive numbers is a square.

1, 1, 11, 57, 31, 20, 25, 50, 64, 464, 1062, 4337, 10091, 21931, 69623, 115913, 227893, 457707, 297126, 1051583, 3377189, 7618873, 12476654, 25832098, 55792448, 75126741, 129180538, 357114149, 823402071, 3902161448, 20043267339, 131420398568

Offset

32,3

Comments

It is unknown whether a circular permutation of the numbers 1 to n exists such that the sum of any two consecutive numbers is a cube.

According to Rivera's Puzzle 311, the smallest n for which a cubic loop exists is 473. - T. D. Noe, Nov 26 2007

From Bert Dobbelaere, Dec 28 2018: (Start)

It is easy to see that no solutions for n <= 30 can exist: for each value of n <= 30 at least one number exists that can only be paired with at most one other number to form a square (e.g., 18 for n=30 can only be paired with 7). No Hamiltonian cycle can exist if the graph contains a vertex of degree less than 2.

For the case n=31, the nonexistence of a Hamiltonian cycle is less trivial but can be shown by hand.

(End)

Links

Table of n, a(n) for n=32..63.

Carlos Rivera, Puzzle 311: Sum to a cube, The Prime Puzzles and Problems Connection.

Formula

a(n) = (A071983(n) - A090460(n)) / (n-1) for n > 1. - Martin Ehrenstein, May 22 2023

Example

There is only one possible square loop of minimum length, which is (32, 4, 21, 28, 8, 1, 15, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31, 5, 11, 25, 24, 12, 13, 3, 6, 30, 19, 17) so a(32)=1.

Crossrefs

Cf. A071983, A090460, A112663.

Sequence in context: A224405 A201150 A114030 * A323039 A211614 A244497

Adjacent sequences: A071981 A071982 A071983 * A071985 A071986 A071987

Keyword

nice,nonn,more,hard

Author

William Rex Marshall, Jun 16 2002

Extensions

a(48)-a(49) from Donovan Johnson, Sep 14 2010

a(50)-a(52) from Giovanni Resta, Nov 11 2012

a(53)-a(54) from Fausto A. C. Cariboni, Sep 19 2018

a(55) from Jud McCranie, Sep 30 2018

a(56) from Jud McCranie, Oct 08 2018

a(57) from Fausto A. C. Cariboni, Oct 24 2018

a(58)-a(61) from Bert Dobbelaere, Dec 28 2018

a(62)-a(63) from Martin Ehrenstein, May 22 2023

Status

approved