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A321727 Number of permutations f of {1,...,n} such that prime(k) + prime(f(k)) + 1 is prime for every k = 1,...,n. 3
1, 1, 1, 2, 6, 10, 31, 76, 696, 4294, 5772, 8472, 128064, 147960, 1684788, 26114739, 523452320, 1029877159, 1772807946, 28736761941, 19795838613, 31445106424, 1313504660737, 54477761675626, 105122845176663, 2200119900732333 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Clearly, a(n) is also the permanent of the matrix of order n whose (i,j)-entry is 1 or 0 according as prime(i) + prime(j) + 1 is prime or not.

Conjecture: a(n) > 0 for all n > 0.

Note that there is no permutation f of {1,...,10} such that prime(k) + prime(f(k)) - 1 is prime for every k = 1,...,10.

LINKS

Table of n, a(n) for n=1..26.

Zhi-Wei Sun, Permutations pi with p_k+p_{pi(k)}+1 prime for all k = 1,...,n, Question 315581 on Mathoverflow, Nov. 17, 2018.

EXAMPLE

a(3) = 1, and (1,2,3) is a permutation of {1,2,3} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(2) + 1 = 7 and prime(3) + prime(3) + 1 = 11 all prime.

a(4) = 2. In fact, (1,2,4,3) is a permutation of {1,2,3,4} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(2) + 1 = 7, prime(3) + prime(4) + 1 = 13 and prime(4) + prime(3) + 1 = 13 all prime; also (1,4,3,2) is a permutation of {1,2,3,4} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(4) + 1 = 11, prime(3) + prime(3) + 1 = 11 and prime(4) + prime(2) + 1 = 11 all prime.

MAPLE

b:= proc(s) option remember; (k-> `if`(k=0, 1, add(`if`(isprime(

      ithprime(i)+ithprime(k)+1), b(s minus {i}), 0), i=s)))(nops(s))

    end:

a:= n-> b({$1..n}):

seq(a(n), n=1..15);  # Alois P. Heinz, Nov 17 2018

MATHEMATICA

p[n_]:=p[n]=Prime[n];

a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[p[i]+p[j]+1]], {i, 1, n}, {j, 1, n}]];

Do[Print[n, " ", a[n]], {n, 1, 22}]

CROSSREFS

Cf. A000040, A321597, A321610, A321611.

Sequence in context: A026117 A283909 A107385 * A145541 A233896 A118039

Adjacent sequences:  A321724 A321725 A321726 * A321728 A321729 A321730

KEYWORD

nonn,more

AUTHOR

Zhi-Wei Sun, Nov 17 2018

EXTENSIONS

a(23)-a(26) from Alois P. Heinz, Nov 17 2018

STATUS

approved

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Last modified October 14 01:36 EDT 2019. Contains 327994 sequences. (Running on oeis4.)