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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)

Original name: Union of nonprimes p_n# < k < p_(n+1)# and gcd(k, p_n#) = 1, with p_n# = A002110(n).

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

Let p_n# = A002110(n). This sequence includes nonprime p_n# < k < p_(n+1)# but does not repeat terms that have already appeared in the sequence (mainly 1 for p_n# with n > 1).

If regarded as a number triangle T(n,k), row length <= A048863(n). (End)

Relevant for sieving primes with a wheel of circumference p#: For the 2*3*5 wheel, the only relevant nonprime residue is 1, while for a 2*3*5*7 wheel, there are 5 more nonprime residues {121, 143, 169, 187, 209}. - M. F. Hasler, Mar 25 2019

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)

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)

It is remarkable that in exponentially increasing ranges these occurrences increase to n=13 and thereafter decline to zero. So A048868 is believed to be finite.

Number of totative pairs (q, k) such that prime q + k nonprime = p_n# and both gcd(q, p_n#) = 1 and gcd(k, p_n#) = 1, with p_n < q <= pi(p_n#), where pi(p_n#) = A000849(n) - n = A048862(n).

Primes p_n < q <= pi(p_n#) are greater than the greatest prime factor of p_n# = p_n, and are thus coprime to p_n#. By the definition of primorial, we need not consider p >= p_n, as these p are divisors of p_n#, i.e., gcd(p, p_n#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (p_n# - q) is not prime in order to count pairs (q, k).

a(n) < floor(A005867(n)/2).

a(n) <= A048862(n).

The totative pair (q,1) = (p_n# - 1, 1) is counted by a(n) for n in A057704, with (p_n# - 1) appearing in A057705.