A108659 -id:A108659 - cp's OEIS frontend

A108660 Square-loop primes.

2, 13, 31, 79, 97, 227, 881, 1013, 2797, 3181, 3631, 8101, 22727, 81001, 101363, 109013, 131363, 181813, 272227, 310181, 310901, 318181, 318881, 631013, 636313, 810401, 818101, 901097, 904097, 972227, 1018813, 1090013, 1810013, 2272727

Offset: 1

Zak Seidov, Jun 16 2005

Primes such that each pair of adjacent digits (and also the first and the last ones) sums up to a square. First term is arguable since there is 'no pair of adjacent digits', but there are the "first" and "last" digits.

Harvey P. Dale, Table of n, a(n) for n = 1..120

Cf. A046704, A007953, A088133, A088134, A088135, A088136, A108659.

Select[Prime[Range[200000]],And@@(IntegerQ[Sqrt[#]]&/@(Total/@Partition[ IntegerDigits[#],2,1,1]))&] (* Harvey P. Dale, Mar 03 2014 *)

A108658 Number of the essentially different permutations of the numbers 0 to n such that the sum of adjacent numbers is a square.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 4, 4, 0, 0, 4, 5, 2, 8, 7, 47, 72, 135, 283, 158, 164, 1948, 1467, 2998, 20561, 66700, 130236, 153058, 181635, 239386, 343189, 1600832, 5001577, 16859525, 45119463, 66785667, 218923884, 393626778, 665307164, 3111228585, 2156371427

Offset: 0

Zak Seidov, T. D. Noe & Max Alekseyev, Jun 16 2005

Square chains (reversals not counted and circles counted once). There is no solution for n=2-13,18-19 (note offset=0). For n=0 and n=1 we have trivial square circles (which are also known as square loops). Square circles seem to appear for all n>30, see A108661. Cf. A090460 for 1-to-n case.

			n=14: one solution
  {8,1,0,9,7,2,14,11,5,4,12,13,3,6,10};
n=15: three solutions
  {0,9,7,2,14,11,5,4,12,13,3,6,10,15,1,8},
  {5,11,14,2,7,9,0,4,12,13,3,6,10,15,1,8},
  {8,1,0,9,7,2,14,11,5,4,12,13,3,6,10,15};
n=16: four solutions
  {0,16,9,7,2,14,11,5,4,12,13,3,6,10,15,1,8},
  {5,11,14,2,7,9,16,0,4,12,13,3,6,10,15,1,8},
  {8,1,0,16,9,7,2,14,11,5,4,12,13,3,6,10,15},
  {8,1,15,10,6,3,13,12,4,5,11,14,2,7,9,0,16}.

Cf. A071984, A090460, A108659, A108660, A108661.

SquareQ[n_]:=IntegerQ[Sqrt[n]]; try[lev_]:=Module[{t, j, circular}, If[lev>n+1, circular=SquareQ[soln[[1]]+soln[[n+1]]]; If[(!circular&&soln[[1]]

a(42)-a(50) from Bert Dobbelaere, Dec 30 2018

A108661 Square loops: the number of circular permutations (reversals not counted as different) of the numbers 0 to n such that the sum of any two consecutive numbers is a square.

Original entry on oeis.org

6, 3, 3, 72, 226, 358, 309, 391, 547, 813, 3562, 10741, 36633, 94547, 120424, 393670, 676579, 1088429, 5531195, 3294327, 8335128, 27820643, 75288569, 111875702, 264015370, 465407197, 687532936, 1109951444, 3256360099

Offset: 31

Zak Seidov, T. D. Noe & Max Alekseyev Jun 16 2005

			There is no solution for n=0,...,30, (note offset=31). For n=0,1 we have the trivial square circles {0} and {0,1}, which are not included in the sequence.
There are only six possible square loops of the minimum length (n=31 case):
{1,0,4,5,31,18,7,29,20,16,9,27,22,3,13,12,24,25,11,14,2,23,26,10,6,30,19,17,8,28,21,15},
{1,0,4,12,13,3,6,30,19,17,8,28,21,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,24},
{1,0,4,21,28,8,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,9,27,22,14,2,23,26,10,15},
{1,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,0,4,21,28,8,17,19,30,6,3,13,12,24},
{1,15,21,28,8,17,19,30,6,10,26,23,2,14,11,5,31,18,7,29,20,16,9,27,22,3,13,12,4,0,25,24},
{1,15,21,28,8,17,19,30,6,10,26,23,2,14,11,25,0,4,5,31,18,7,29,20,16,9,27,22,3,13,12,24}.
In the n=32,33 (resp.) cases, there are three square loop solutions:
{1,0,4,32,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,9,27,22,14,2,23,26,10,15,21,28,8},
{1,8,28,21,4,32,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,0,9,27,22,14,2,23,26,10,15},
{1,8,28,21,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,0,4,32,17,19,30,6,3,13,12,24},
and
{1,0,4,32,17,19,30,6,3,13,12,24,25,11,5,20,29,7,18,31,33,16,9,27,22,14,2,23,26,10,15,21,28,8},
{1,8,28,21,4,32,17,19,30,6,3,13,12,24,25,11,5,20,29,7,18,31,33,16,0,9,27,22,14,2,23,26,10,15},
{1,8,28,21,15,10,26,23,2,14,22,27,9,16,33,31,18,7,29,20,5,11,25,0,4,32,17,19,30,6,3,13,12,24}
(resp.).

Cf. A108658 = square chains.

Cf. A071984, A090460, A108658, A108659, A108660.

a(42)-a(47) from Donovan Johnson, Sep 14 2010
a(48)-a(52) from Fausto A. C. Cariboni, Sep 21 2018
a(53)-a(59) from Bert Dobbelaere, Dec 29 2018

cp's OEIS Frontend

A108660 Square-loop primes.

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A108658 Number of the essentially different permutations of the numbers 0 to n such that the sum of adjacent numbers is a square.

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Author

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Programs

Mathematica

Extensions

A108661 Square loops: the number of circular permutations (reversals not counted as different) of the numbers 0 to n such that the sum of any two consecutive numbers is a square.

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Author

Keywords

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