A321805
Number of permutations f of {1,...,n} such that k!*f(k) + 1 is prime for every k from 1 to n.
Original entry on oeis.org
1, 2, 4, 6, 10, 10, 13, 40, 212, 702, 3531, 19008, 34858, 39764, 102312, 47927, 94860, 232006, 658766, 829583, 1547703, 2040211, 32073218, 51347260, 496226762, 1504307318, 16663026685, 125080784519, 241032642271, 1216752358950, 2147004248698, 9320087810948, 19383919945950, 16259146126113, 81023699301023, 124167501991213
Offset: 1
a(2) = 2 since (1,2) and (2,1) are permutations of {1,2} with 1!*1 + 1 = 2, 2!*2 + 1 = 5, 1!*2 + 1 = 3 and 2!*1 + 1 = 3 all prime.
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a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[i!*j+1]],{i,1,n},{j,1,n}]]; Do[Print[n," ",a[n]],{n,1,27}]
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a(n)={matpermanent(matrix(n, n, i, j, isprime(i!*j+1)))} \\ Andrew Howroyd, Nov 19 2018
A321855
Number of permutations f of {1,...,n} such that prime(k)*prime(f(k)) - 2 is prime for every k = 1,...,n.
Original entry on oeis.org
1, 1, 2, 3, 5, 12, 2, 3, 65, 248, 448, 1792, 4288, 6468, 27068, 29752, 106066, 447982, 1250762, 6304196, 46613084, 126391780, 504582496, 2270372946, 3028652541, 8941959118, 36442298864, 175008626450, 318369805106, 1974700703920, 6654020288821, 48819526290634, 150577775767875, 574885284627624, 3058310882340228, 15949743649457780
Offset: 1
a(7) = 2. The only even permutation of {1,...,7} meeting the requirement is (1,5,7,4,2,6,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(4) - 2 = 47, prime(5)*prime(2) - 2 = 31, prime(6)*prime(6) - 2 = 167 and prime(7)*prime(3) - 2 = 83 all prime. Also, the only odd permutation of {1,...,7} meeting the requirement is (1,5,7,6,2,4,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(6) - 2 = 89, prime(5)*prime(2) - 2 = 31, prime(6)*prime(4) - 2 = 89 and prime(7)*prime(3) - 2 = 83 all prime.
- Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), pp. 157-176.
- Zhi-Wei Sun, Chen primes and permutations, Question 315679 on Mathoverflow, Nov. 19, 2018.
- Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.
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Permanent[m_List]:=With[{v = Array[x, Length[m]]},Coefficient[Times @@ (m.v), Times @@ v]];
a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[Prime[i]*Prime[j]-2]],{i,1,n},{j,1,n}]];
Do[Print[n," ",a[n]],{n,1,27}]
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a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)*prime(j) - 2))); \\ Jinyuan Wang, Jun 13 2020
A321766
Number of permutations f of {1,...,n} such that 3^k + 3^(f(k)) - 1 is prime for every k = 1,...,n.
Original entry on oeis.org
1, 2, 3, 11, 14, 33, 59, 290, 843, 690, 231, 978, 2896, 2966, 38252, 384917, 22351, 68546, 28245, 147459, 84578, 17647, 17647, 232213, 17647, 792, 93640, 785178, 5635699, 11658279, 67706584, 351837312, 233636388, 26967286, 35027435, 242576452
Offset: 1
a(2) = 2 since both (1,2) and (2,1) are permutations of {1,2}, and 3^1 + 3^1 - 1 = 5, 3^2 + 3^2 - 1 = 17, 3^1 + 3^2 - 1 = 11 and 3^2 + 3^1 - 1 = 11 are all prime.
- Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)
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N:= 25: # to get a(1)..a(N)
q:= proc(i,j) if isprime(3^i+3^j-1) then 1 else 0 fi end proc:
M:= Matrix(N,N, q, shape=symmetric):
seq(LinearAlgebra:-Permanent(M[1..n,1..n]), n=1..N); # Robert Israel, Dec 08 2019
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a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[3^i+3^j-1]],{i,1,n},{j,1,n}]];
Do[Print[n," ",a[n]],{n,1,30}]
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a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(3^i + 3^j - 1))); \\ Jinyuan Wang, Jun 13 2020
Showing 1-3 of 3 results.
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