A321805 -id:A321805 - cp's OEIS frontend

A335233 Numbers m such that m!*i + 1 is composite for i = 1..m.

50, 66, 67, 76, 81, 110, 117, 120, 122, 157, 178, 183, 217, 219, 226, 235, 240, 242, 244, 250, 254, 260, 266, 269, 274, 275, 287, 289, 309, 326, 346, 354, 363, 379, 380, 386, 400, 407, 410, 417, 419, 443, 449, 451, 470, 474, 489, 493, 509, 513, 518, 541, 543

Offset: 1

Chai Wah Wu, Jun 09 2020

nonn

If m is a term, then A321805(m) = 0.

Chai Wah Wu, Table of n, a(n) for n = 1..240

Cf. A321805.

Select[Range@ 200, (f = #!; NoneTrue[f*Range[#] + 1, PrimeQ]) &] (* Robert Price, Sep 14 2020 *)

from sympy import isprime
A335233_list, f = [], 1
for k in range(1,100):
    f *= k
    g = 1
    for i in range(1,k+1):
        g += f
        if isprime(g):
            break
    else:
        A335233_list.append(k)

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

Zhi-Wei Sun, Nov 19 2018

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for each n > 0, there is an even permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n. Also, for any integer n > 2, there is an odd permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n.
If we let b(n) denote the number of even permutations f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n, then (b(1),...,b(11)) = (1,1,1,1,3,6,1,1,33,125,226).
In 1973 J.-R. Chen proved that there are infinitely many primes p with p + 2 a product of at most two primes, such primes p are now called Chen primes.

			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.

Cf. A000040, A109611, A257926, A321597, A321610, A321611, A321727, A321805.

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}]

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)*prime(j) - 2))); \\ Jinyuan Wang, Jun 13 2020

a(28)-a(29) from Jinyuan Wang, Jun 13 2020

a(30)-a(36) from Vaclav Kotesovec, Aug 20 2021

A335361 Prime numbers p such that p!*i + 1 is composite for i = 1..p.

Original entry on oeis.org

67, 157, 269, 379, 419, 443, 449, 509, 541, 577, 743, 769, 859, 863, 929, 937, 1009, 1087, 1163, 1213, 1217, 1367, 1381, 1481, 1579, 1733, 1747, 1753, 1783, 1787, 1877, 1901, 1997, 2153

Offset: 1

Chai Wah Wu, Jun 10 2020

Primes in A335233.

Cf. A035093, A321805, A335233.

Select[Prime@ Range@ 100, (f = #!; NoneTrue[f*Range[#] + 1, PrimeQ ]) &] (* Robert Price, Sep 14 2020 *)

is(p) = if(isprime(p), for(i=1, p, if(ispseudoprime(i*p!+1), return(0))); 1, 0); \\ Jinyuan Wang, Jun 21 2020

from sympy import isprime, nextprime, factorial
A335361_list, p = [], 2
while p < 500:
    f, g = factorial(p), 1
    for i in range(1,p+1):
        g += f
        if isprime(g):
            break
    else:
        A335361_list.append(p)
    p = nextprime(p)

a(34) from Jinyuan Wang, Jun 21 2020

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A335233 Numbers m such that m!*i + 1 is composite for i = 1..m.

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A321855 Number of permutations f of {1,...,n} such that prime(k)*prime(f(k)) - 2 is prime for every k = 1,...,n.

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A335361 Prime numbers p such that p!*i + 1 is composite for i = 1..p.

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PARI

Python

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