A036440 Number of ways of arranging row n of the prime pyramid.

1, 1, 1, 1, 1, 1, 2, 4, 7, 24, 80, 216, 648, 1304, 3392, 13808, 59448, 155464, 480728, 1588162, 5626309, 28279112, 157469880, 842498189, 4998554801, 28466978744, 166572523589, 1020487969891, 5753385660978

Offset: 1

Author

John W. Layman

Keywords

• nonn
• nice

Comments

Number of ways to arrange numbers from 1 to n in a row, starting with 1 and ending with n, such that the sum of every two adjacent numbers is prime.

From Daniel Forgues, May 18 2011: (Start)

Since the sum of any two adjacent entries is at least 3, the sum is thus an odd prime, which implies that any two consecutive entries have opposite parity. Since the first and last entries of row n are fixed to 1 and n, we have to find n-2 entries, where ceiling((n-2)/2) are even and floor((n-2)/2) are odd, so for row n the number of arrangements to investigate is

(ceiling((n-2)/2))! * (floor((n-2)/2))! (Cf. A010551(n-2), n >= 2.)

Prime pyramids are also (more fittingly?) called prime triangles. (End)

Examples

			a(8)=4 because of the 720 permutations P of {1,2,3,4,5,6,7,8} with first element 1 and last element 8, these four satisfy the "prime pyramid" condition that P[i] + P[i+1] be prime for i=1..7:
  1 2 3 4 7 6 5 8; (lexicographically earliest row 8)
  1 2 5 6 7 4 3 8;
  1 4 7 6 5 2 3 8;
  1 6 7 4 3 2 5 8.
For row 8, there are 6! = 720 permutations of {2,3,4,5,6}, but if we take into account that the parity of all entries of row n must alternate, we only have to consider ceiling((8-2)/2)! * floor((8-2)/2)! = 36 cases.
		

References

• R. K. Guy, Unsolved Problems Number Theory, C1.

Links

• Eric Weisstein's World of Mathematics, Prime Triangle.

Crossrefs

Cf. A051237 for lexicographically earliest prime pyramid.

Programs

Mathematica

prsQ[n_]:=AllTrue[Total/@Partition[n,2,1],PrimeQ]; Table[Count[Join[{1},#,{n}]&/@ Permutations[ Range[2,n-1]],?prsQ],{n,12}] (* _Harvey P. Dale, May 07 2023 *)

Extensions

More terms from Jud McCranie

a(25)-a(27) from Max Alekseyev, Jan 05 2008

a(28)-a(29) from Giovanni Resta, Apr 01 2014