A345124 a(n) is the smallest k such that f(k) is composite for all m-fold compositions f of the functions 6*x +- 1, 1 <= m <= n.

20, 50, 284, 1868, 47951, 6245927, 15932178151

Offset: 1

Author

Marc Morgenegg, Oct 06 2021

Keywords

• nonn
• more

Comments

Proof that a(n) exists for all n: The numbers that are equal to f(k) for some m-fold composition f of the functions 6*x +- 1 can be written as 6^m*k +- c, where c is in the set C_m, defined by C_1 = {1} and C_{m+1} = {6*c +- 1 for c in C_m}. Choose a positive integer k_0 that is divisible by all numbers in C_m for 1 <= m <= n. Then 6^m*k_0 +- c is divisible by (and greater than) c, so it is composite if c > 1. (In fact, the largest number in C_m is A003464(m).) Since there are arbitrarily long prime gaps, we can choose a positive integer r such that 6*k_0*r +- 1 are both composite. With k = k_0*r, the numbers 6^m*k +- c will all be composite for c in C_m, 1 <= m <= n, as desired. - Pontus von Brömssen, Nov 01 2021

Examples

			Formula for the twin composites by iteration n:
n=1: 6*k+-1.
n=2: 6*(6*k+-1)+-1.
n=3: 6*(6*(6*k+-1)+-1)+-1.
Term a(n) example for smallest number k for iteration n:
a(1)=20, 6*20-1=119, 6*20+1=121, all {119,121} are composite numbers.
a(2)=50, 6*50-1=299, 6*50+1=301, 6*(6*50-1)-1=1793, 6*(6*50-1)+1=1795, 6*(6*50+1)-1=1805, 6*(6*50+1)+1=1807, all {299,301,1793,1795,1805,1807} are composite numbers.
		

Crossrefs

Cf. A003464, A060461 (numbers k such that 6*k+-1 are twin composites).

Programs

Mathematica

a[n_] := Module[{k = 1}, While[!AllTrue[Flatten@ Rest@ NestList[Flatten@ Join[{6*# - 1, 6*# + 1}] &, k, n], CompositeQ], k++]; k]; Array[a, 5] (* Amiram Eldar, Oct 25 2021 *)

Python

from sympy import isprime
def A345124(n):
    C = [[1]]
    for i in range(n-1):
        C.append(sum(([6*c-1,6*c+1] for c in C[-1]),[]))
    k = 1
    while 1:
        k6 = 6*k
        for i in range(n):
            if any(isprime(k6-c) or isprime(k6+c) for c in C[i]):
                break
            k6 *= 6
        else:
            return k
        k += 1 # Pontus von Brömssen, Nov 01 2021

Extensions

More terms from Pontus von Brömssen, Oct 06 2021

Name edited by Pontus von Brömssen, Nov 01 2021

a(7) from Martin Ehrenstein, Nov 13 2021