cp's OEIS Frontend

Numbers n such that gcd(n, 2^n-2) > 1 and gcd(n, b^n-b) = 1 for some b > 2, b < n.

Or: Numbers n such that gcd(n, 2^n-2) > 1 and for every prime factor p of n, p-1 does not divide n-1.

2057 is the first term not in A008367, nor in A287391. - M. F. Hasler, Oct 04 2018

Composite numbers k such that k^720 mod 30030 = 1. - Gary Detlefs, May 02 2012

The asymptotic density of this sequence is 192/1001. - Amiram Eldar, Mar 22 2021

Terms of A285784 that have a(n) = 1 appear in A287390.

Terms of A285784 that have a(n) > 1 appear in A287391.

From Michael De Vlieger, Jun 09 2017: (Start)

Let primorial p_n# = A002110(n) and let m be a nonzero positive number called a totative such that gcd(t, p_n#) = 1. This sequence concerns nonprime m. A285784 is the sequence that lists unique nonprime totatives m of primorials p_n#.

For A285784(1), a(n) = infinity, since 1 is the empty product and a totative of (i.e., coprime to) all numbers. Hence the offset of a(n) is 2 and for this reason hereinafter we only consider composite totatives m.

Consider the composite totative m in A285784. For a given composite term in A285784, there is a least primorial p_a# to which m is coprime. Such m < p_a# are products of prime totatives q > p_a, the gpf of p_a#. Therefore m "appears" when there are prime totatives q < sqrt(p_a#). The smallest a for which we have this condition is a = 4, as q = 11 is less than sqrt(210). For the same reason the first composite term of A285784 is 11^2 = 121.

For n >= 2, m is coprime to a finite range of primorials p_a# .. p_b#. If m is coprime to p_b#, then it must be coprime to all primorials p_a# .. p_b# by the definition of primorial. m is no longer coprime to p_(b+1)# since at least one of its prime divisors p_(b+1) also divides p_(b+1)#. This sequence gives the range b - a + 1.

To generate data that includes all the terms of A285784 less than a limit x, we can write a while statement that operates so long as there is at least 1 totative m < x of p_n#. Since primorial p_n# is the product of the smallest n primes, fewer numbers less than x are coprime to p_n# as n increases, until exhaustion. Thus we can produce a list of unique m < x (i.e., terms of A285784 less than x) for relatively large primorials p_n#. Then we can count the instances of terms of A285784 for a list of lists of totatives m < x for primorials p_1# .. p_n# and obtain certainty about the number of instances of terms of A285784.

First position of values of a(n): {2, 4, 12, 20, 38, 47, 76, 96, 111, 139, 228, 241, 300, 339, 363, 434, 482, 566, 689, 752, 790, 862, 902, 973, 1264, 1361, 1506, 1562, 1816, ...}

Terms of A285784 that set records in a(n): {121, 169, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, ...}

(End)