A356187 - OEIS

%I #19 Aug 06 2022 08:07:22

%S 1,1,0,0,0,2,2,6,4,24,6,162,330,1428,1632

%N Number of permutations f of {1,...,n} with f(1) = 1 such that those k*f(k) + 1 (k = 1..n) are n distinct primes.

%C Conjecture: a(n) > 0 except for n = 3,4,5. Also, for any integer n > 5, there is a permutation f of {1,...,n} with f(1) = 3 such that those k*f(k) - 1 (k = 1..n) are n distinct primes.

%C This is stronger than part (i) of the conjecture in A321597.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/s10801-021-01028-8">On permutations of {1,...,n} and related topics</a>, J. Algebraic Combin., 2021.

%e a(7) = 2 since the only permutations f of {1,...,7} with f(1) = 1 such that {k*f(k) + 1: k = 1..7} is a set of 7 primes, are (f(1),...,f(7)) = (1,3,4,7,2,5,6) and (1,5,2,3,6,7,4). Note that 1*1 + 1 = 2, 2*3 + 1 = 7, 3*4 + 1 = 13, 4*7 + 1 = 29, 5*2 + 1 = 11, 6*5 + 1 = 31, 7*6+1 = 43 are distinct primes. Also, 1*1 + 1 = 2, 2*5 + 1 = 11, 3*2 + 1 = 7, 4*3 + 1 = 13, 5*6 + 1 = 31, 6*7 + 1 = 43, 7*4 + 1 = 29 are distinct primes.

%e a(10) > 0 since for (f(1),...,f(10)) = (1,3,4,7,8,5,6,9,2,10) the set {k*f(k) + 1: k = 1..10} is a set of 10 distinct primes.

%t (* A program to find all the permutations f of {1,...,9} with f(1) = 1 such that U = {k*f(k)+1: k = 1..9} is a set of 9 distinct primes. *)

%t V[i_]:=V[i]=Part[Permutations[{2,3,4,5,6,7,8,9}],i]

%t m=0;Do[U={2};Do[p=j*V[i][[j-1]]+1;If[PrimeQ[p],U=Append[U,p]],{j,2,9}];

%t If[Length[Union[U]]==9,m=m+1;Print[m," ",V[i]," ",U]],{i,1,8!}]

%o (Python)

%o from itertools import permutations as perm

%o from itertools import islice

%o from sympy import isprime

%o from math import factorial as fact

%o import collections

%o def consume(iterator, n=None):

%o     "Advance the iterator n-steps ahead. If n is None, consume entirely."

%o     # Use functions that consume iterators at C speed.

%o     if n is None:

%o         # feed the entire iterator into a zero-length deque

%o         collections.deque(iterator, maxlen=0)

%o     else:

%o         # advance to the empty slice starting at position n

%o         next(islice(iterator, n, n), None)

%o for x in range(2,20):

%o     mult = range(1,x)

%o     count = 0

%o     q = perm(range(1,x))

%o     for y in q:

%o         keeppos = 0

%o         keepflag = False

%o         if y[0] != 1:#stop when the first digit is not 1

%o             break

%o         z = [mult[a] * y[a] + 1 for a in range(x-1)]

%o         for b in z[0:-2]:

%o             if not isprime(b):

%o                 keeppos = z.index(b)

%o                 keepflag = True

%o                 break

%o         if keepflag:#skip ahead to advance the next non-prime term

%o             consume(q,fact(x-keeppos-2)-1)

%o         elif len(set(z)) == len(z) and all(isprime(b) for b in set(z)):#no duplicates and all primes

%o             count += 1

%o     print(x-1,count)

%o # _David Consiglio, Jr._, Aug 04 2022

%Y Cf. A000040, A321597, A333083.

%K nonn,more

%O 1,6

%A _Zhi-Wei Sun_, Jul 28 2022

%E a(11)-a(13) from _Jinyuan Wang_, Jul 29 2022

%E a(14)-a(15) from _David Consiglio, Jr._, Aug 04 2022