A321610
Number of permutations tau of {1,...,n} such that k^2 + tau(k)^2 is prime for every k = 1,...,n.
Original entry on oeis.org
1, 1, 1, 1, 1, 4, 0, 16, 4, 144, 64, 81, 256, 5184, 1600, 25600, 8100, 183184, 108900, 5924356, 342225, 9066121, 11356900, 106853569, 105698961, 16119349444, 1419933124, 69792129124, 14251584400, 613950602500, 304388337796, 25042678198756, 10080904401936, 1179245283899881, 1045903153861476, 31082438574307129
Offset: 1
a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13 and 3^2 + 2^2 all prime.
a(5) = 1, and (1,3,2,5,4) is a permutation of {1,2,3,4,5} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13, 3^2 + 2^2 = 13, 4^2 + 5^2 = 41 and 5^2 + 4^2 = 41 all prime.
-
V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]
Do[r=0;Do[Do[If[PrimeQ[i^2+Part[V[n],k][[i]]^2]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]
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a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i^2 + j^2))); \\ Jinyuan Wang, Jun 13 2020
A321611
Number of permutations tau of {1,...,n} such that k^4 + tau(k)^4 is prime for every k = 1,...,n.
Original entry on oeis.org
1, 1, 1, 4, 4, 4, 4, 64, 16, 144, 144, 0, 144, 144, 289, 4356, 2916, 22500, 79524, 1887876, 313600, 3459600, 2985984, 50069776, 32353344, 2056803904, 237591396, 11713732900, 10265337124, 342040164964, 30744816964, 2507750953744, 378640854244, 53517915572836, 7415600385600, 230030730231696
Offset: 1
a(3) = 1, and (1,3,2) is a permutation of {1,...,n} with 1^4 + 1^4 = 2, 2^4 + 3^4 = 97 and 3^4 + 2^4 = 97 all prime.
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V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]
Do[r=0;Do[Do[If[PrimeQ[i^4+Part[V[n],k][[i]]^4]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]
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a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i^4 + j^4))); \\ Jinyuan Wang, Jun 13 2020
A321727
Number of permutations f of {1,...,n} such that prime(k) + prime(f(k)) + 1 is prime for every k = 1,...,n.
Original entry on oeis.org
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, 2761739099984389, 83123428119278837, 219734505495953342, 7228968492870136475, 13623311188546432233, 625620139149376861330, 18603738861035365389401, 64952397216275572992159, 3115094155636931821691880, 4788927142804364353625983
Offset: 1
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.
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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
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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}]
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a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)+prime(j)+1))); \\ Jinyuan Wang, Jun 13 2020
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.
-
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
A356187
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.
Original entry on oeis.org
1, 1, 0, 0, 0, 2, 2, 6, 4, 24, 6, 162, 330, 1428, 1632
Offset: 1
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.
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.
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(* 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. *)
V[i_]:=V[i]=Part[Permutations[{2,3,4,5,6,7,8,9}],i]
m=0;Do[U={2};Do[p=j*V[i][[j-1]]+1;If[PrimeQ[p],U=Append[U,p]],{j,2,9}];
If[Length[Union[U]]==9,m=m+1;Print[m," ",V[i]," ",U]],{i,1,8!}]
-
from itertools import permutations as perm
from itertools import islice
from sympy import isprime
from math import factorial as fact
import collections
def consume(iterator, n=None):
"Advance the iterator n-steps ahead. If n is None, consume entirely."
# Use functions that consume iterators at C speed.
if n is None:
# feed the entire iterator into a zero-length deque
collections.deque(iterator, maxlen=0)
else:
# advance to the empty slice starting at position n
next(islice(iterator, n, n), None)
for x in range(2,20):
mult = range(1,x)
count = 0
q = perm(range(1,x))
for y in q:
keeppos = 0
keepflag = False
if y[0] != 1:#stop when the first digit is not 1
break
z = [mult[a] * y[a] + 1 for a in range(x-1)]
for b in z[0:-2]:
if not isprime(b):
keeppos = z.index(b)
keepflag = True
break
if keepflag:#skip ahead to advance the next non-prime term
consume(q,fact(x-keeppos-2)-1)
elif len(set(z)) == len(z) and all(isprime(b) for b in set(z)):#no duplicates and all primes
count += 1
print(x-1,count)
# David Consiglio, Jr., Aug 04 2022
A321651
Number of even permutations f of {1,...,n} such that k^3 + f(k)^3 is a practical number for every k = 1,...,n.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 6, 24, 36, 180, 840
Offset: 1
a(5) = 1, and (5,4,3,2,1) is an even permutation of {1,2,3,4,5} with 1^3 + 5^3 = 126, 2^3 + 4^3 = 72, 3^3 + 3^3 = 54, 4^3 + 2^3 = 72 and 5^3 + 1^3 = 126 all practical.
- Paul Bradley, Prime number sums, arXiv:1809.01012 [math.GR], 2018.
- Zhi-Wei Sun, Primes arising from permutations, Question 315259 on Mathoverflow, Nov. 14, 2018.
- Zhi-Wei Sun, Primes arising from permutations (II), Question 315341 on Mathoverflow, Nov. 14, 2018.
- Zhi-Wei Sun, A mysterious connection between primes and squares, Question 315351 on Mathoverflow, Nov. 15, 2018.
-
f[n_]:=f[n]=FactorInteger[n];
Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]];
Do[r=0;Do[If[Signature[Part[V[n],k]]==-1,Goto[aa]];Do[If[pr[i^3+Part[V[n],k][[i]]^3]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]
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
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