1, 2, 4, 48, 384, 2304, 4608, 18432, 552960, 19906560, 796262400, 33443020800, 66886041600, 267544166400, 535088332800, 32105299968000, 2118949797888000, 148326485852160000, 10679506981355520000, 833001544545730560000, 1666003089091461120000, 146608271840048578560000
Offset: 3
First example:
Let R = {1, 7, 11, 13, 17, 19, 23, 29} be the set of the multiplicative group of integers modulo 30 (or it is the set of positive integers less than 2*3*5 and prime to 2*3*5, or it is the reduced residue system for the 3rd primorial number 30 = A002110(3)).
The cardinality of |R| is equal to 8 elements, 8 = A005867(3).
The set {1} is the shortcut set of the multiplicative group of integers modulo 30.
The cardinality of |{1}| = 1, i.e., a(1) = 1.
R = { (30/2 +- {1}*2^{1, ..., 4}) mod 30 } = {(30/2 +- {1}*2^{1,2,3,4}) mod 30}={(15+2^1) mod 30, (15-2^1) mod 30,(15+2^2) mod 30, (15-2^2) mod 30, (15+2^3) mod 30, (15-2^3) mod 30, (15+2^4) mod 30, (15-2^4) mod 30} = {1,7,11,13,17,19,23,29},
where 30 = A002110(3), 4 = A155747(2), |R| = A005867(3).
4 is the smallest number m with the property that 2^m-1 is divisible by the first n odd primes.
Second example:
Let R = {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209} be the set of the multiplicative group of integers modulo 210 (or it is the set of positive integers less than 2*3*5*7 and prime to 2*3*5*7, or it is the reduced residue system for the 4th primorial number 210 = A002110(4)).
The cardinality of |R| is equal to 48 elements, 48 = A005867(5).
The set {1, 11} is the shortcut set of the multiplicative group of integers modulo 210.
The cardinality of |{1, 11}| = 2, i.e., a(2) = 2.
R = { ( 210/2 +- {1, 11}*2 ^{1..12}) mod 210 }, where 210 = A002110(4), 12 = A155747(3), |R| = A005867(4).
12 is the smallest number m with the property that 2^m-1 is divisible by the first n odd primes.
General case:
R = {R(1), ..., R(phi(prime(n)#))},
R = {(prime(n)#/2 +- {R(1),R(2),...,R(k)}*2^{1,2,...,z}) mod prime(n)#}, where prime(n)# is the product of the first n primes, i.e., it is A002110(n); phi is Euler's totient function, which counts the positive integers up to a given argument that are relatively prime to the argument, in our case phi(prime(n)#) is A005867;
R is the set of the multiplicative group of integers modulo prime(n)#;
z is a term of A155747.
k is a term of a(n).
R(1) <= R(k) < R(phi(prime(n)#)).
Table:
+-----+-----------+----------------+---------+------+
| n | A002110 | A005867 | A155747 | a(n) |
| | prime(n)# | phi(prime(n)#) | z | k |
+-----+-----------+----------------+---------+------+
| 0 | 1 | 1 | - | - |
| 1 | 2 | 1 | - | - |
| 2 | 6 | 2 | 2 | - |
| 3 | 30 | 8 | 4 | 1 |
| 4 | 210 | 48 | 12 | 2 |
| 5 | 2310 | 480 | 60 | 4 |
| 6 | 30030 | 5760 | 60 | 48 |
| 7 | 510510 | 92160 | 120 | 384 |
| 8 | 9699690 | 1658880 | 360 | 2304 |
| .. | ... | ... | ... | ... |
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