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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)

Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.

From Michael De Vlieger, May 18 2017: (Start)

Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).

Numbers in the reduced residue system of A002110(n).

A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.

A048862(n) = number of primes in row n of a(n).

A048863(n) = number of nonprimes in row n of a(n).

Since 1 is coprime to all n, it delimits the rows of a(n).

The prime A000040(n+1) is the second term in row n since it is the smallest prime coprime to A002110(n) by definition of primorial.

The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.

The Kummer numbers A057588(n) = A002110(n) - 1 are the last terms of rows n, since (n - 1) is less than and coprime to all positive n. (End)

From Michael De Vlieger, May 24 2017; corrected and edited by M. F. Hasler, Oct 04 2018: (Start)

Let p_n# = A002110(n). This sequence lists 1 and composite numbers p_n# < k < p_(n+1)# for all positive n such that least_prime_factor(k) > p_(n+2).

Subset of A285784.

If considered as an irregular number triangle T(n,k), row lengths n < A048863(n).

(End)

Only these 6 values are not prime numbers up to n=499: 1, 590221, 2807627, 5862793, 39109337, 13116283.

All a(n) are totatives of A002110(n); thus if a(n) < b^2 in the semiprime b*c then a(n) is prime, otherwise a(n) is either prime or semiprime.

The number c is prevprime(p_n# / p_(n+1)), where p_n# = A002110(n). Thus semiprime b*c = A000040(n+1)*prevprime(A002110(n) / A000040(n+1)), and a(n) = A002110(n) - A000040(n+1)*prevprime(A002110(n)/A000040(n+1)). - Michael De Vlieger, May 15 2017

From Michael De Vlieger, May 24 2017: (Start)

Let p_n# = A002110(n). Composite numbers p_n# < k < p_(n+1)# such that gcd(k,p_(n+1)) = 1 and whose minimum prime divisor is p_(n+2).

Subsequence of A285784.

The sequence can be thought of as an irregular triangle T(n,k) with the first terms appearing for n = 3. Row lengths of T(n,k) < A048863(n).

Many of the terms are semiprimes p_(n+2)*q with p_(n+2) < q < p_pi(p_(n+1)#), where pi(x) = A000720(x).

The smallest square in a(n) is 121 = 11^2. The smallest p^m for m = {2, 3, 4, 5} is {121, 2197, 130321, 643343}, which are {11^2, 13^3, 19^4, 23^5} respectively.

(End)

List of nonprime totatives t of m for m in A036913.

Nonprime 1 is coprime to all numbers, thus a(1) = 1.

The integers {175, 245, 275} are absent, distinguishing this sequence from A038509 and A067793. These terms have factors 5^2 * 7, 5 * 7^2, 5^2 * 11. Only the terms in positions {2, 3, 4, 6, 8, 11, 18} of A036913 (i.e., {6, 12, 18, 42, 66, 126, 462}) are larger and coprime to 5. Of these only 462 is greater than these three terms, however 462 is divisible by 7 and 11. Thus {175, 245, 275} are not terms.

Squared primes q^2 for q >= 5 appear in the sequence at positions {2, 4, 13, 20, 35, 48, 71, 107, 123, 173, ...}. These are coprime to and smaller than {42, 60, 126, 210, 330, 420, ...} at indices {6, 7, 11, 13, 16, 17, 20, 25, 25, 28, 30, 30, 31, 40, 33, 35, ...} in A036913.