A073452 -id:A073452 - cp's OEIS frontend

A073451 Number of essentially different ways in which the squares 1,4,9,...,n^2 can be arranged in a sequence such that all pairs of adjacent squares sum to a prime number. Rotations and reversals are counted only once.

1, 1, 1, 1, 2, 4, 0, 12, 6, 66, 156, 44, 312, 1484, 2672, 6680, 19080, 45024, 168496, 2033271, 724543, 2776536, 24598062, 26849699, 345160845, 4478968678, 5094833662, 14184530127, 29116554754, 125878922175

Offset: 1

T. D. Noe, Aug 02 2002

Note that when the first and last numbers of an arrangement sum to a prime, then there are n rotations that are treated as one arrangement. The case n=10 exhibits the first of these rotational solutions: {1,4,9,64,49,100,81,16,25,36}. The Mathematica program uses a backtracking algorithm to count the arrangements. To print the unique arrangements, remove the comments from around the print statement.

			a(5)=2 because there are two essential different arrangements: {9,4,1,16,25} and {9,4,25,16,1}.

Carlos Rivera, Puzzle 189: Squares and primes in a row, The Prime Puzzles & Problems Connection.

Cf. A072129, A073452.

nMax=12; $RecursionLimit=500; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=PrimeQ[soln[[1]]^2+soln[[n]]^2]; If[(!circular&&soln[[1]]

a(24)-a(30) from Martin Ehrenstein, Jul 19 2023

A182540 Number of ways of arranging the numbers 1 through n on a circle so that no sum of two adjacent numbers is prime, up to rotations and reflections.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 44, 208, 912, 8016, 61952, 671248, 8160620, 87412258, 888954284, 12156253488, 180955852060, 2907927356451, 50317255621843, 802326797235038, 12251146829850324, 233309934271940028, 4243527581615332664, 79533825261873435894, 1602629887788636447221, 30450585799991840921483, 622433536382831426225696, 14891218890120375419560713, 344515231090957672408413959

Offset: 1

Jens Voß, May 04 2012

nonn

			If n < 6, then in every arrangement of the numbers 1 through n on a circle, there are two adjacent numbers adding up to a prime. For n = 6, the only arrangement without a prime sum is (1, 3, 6, 2, 4, 5).

Cf. A051252, A073452, A191374

a(15)-a(17) from Alois P. Heinz, May 04 2012
a(18) from R. H. Hardin, May 07 2012
a(19)-a(30) from Max Alekseyev, Aug 19 2013

A074063 a(n) is the number of essentially different ways in which the integers 1,2,3,...,n can be arranged in a sequence such that (1) adjacent integers sum to a prime number and (2) squares of adjacent numbers sum to a prime number. Rotations and reversals are counted only once.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 0, 0, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1481, 4266, 0, 0, 5624, 0

Offset: 1

T. D. Noe, Aug 17 2002

It is known that a(n) > 0 for 58 <= n <= 200. It is conjectured that a(n) > 0 for all n > 57. A greedy algorithm can be used to quickly find a solution for many n. See the link to puzzle 189 for more details. The Mathematica program uses a backtracking algorithm to count the arrangements. To print the unique arrangements, remove the comments from around the print statement.

a(58) > 2.65*10^6; a(59) > 4.45*10^6. - Alexander D. Healy, Apr 07 2025

			a(4)=1 because there is essentially one arrangement: {3,2,1,4}.

Carlos Rivera, Puzzle 189: Squares and primes in a row, The Prime Puzzles & Problems Connection.

Cf. A073451, A073452.

nMax=12; $RecursionLimit=500; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=PrimeQ[soln[[1]]^2+soln[[n]]^2]&&PrimeQ[soln[[1]]+soln[[n]]]; If[(!circular&&soln[[1]]

a(52)-a(57) from Alexander D. Healy, Apr 01 2025

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A073451 Number of essentially different ways in which the squares 1,4,9,...,n^2 can be arranged in a sequence such that all pairs of adjacent squares sum to a prime number. Rotations and reversals are counted only once.

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A182540 Number of ways of arranging the numbers 1 through n on a circle so that no sum of two adjacent numbers is prime, up to rotations and reflections.

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A074063 a(n) is the number of essentially different ways in which the integers 1,2,3,...,n can be arranged in a sequence such that (1) adjacent integers sum to a prime number and (2) squares of adjacent numbers sum to a prime number. Rotations and reversals are counted only once.

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Mathematica

Extensions