This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A286424 #10 Feb 16 2025 08:33:45
%S A286424 0,0,1,1,4,110,1432,23338,397661,8193828,212858328,5941706227
%N A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).
%C A286424 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).
%C A286424 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).
%C A286424 a(n) < floor(A005867(n)/2).
%C A286424 a(n) <= A048862(n).
%C A286424 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.
%H A286424 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Primorial">Primorial</a>.
%H A286424 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Totative.html">Totative</a>.
%F A286424 a(n) = (A000010(A002110(n)) - A048863(n)) - 2*A117929(A002110(n))
%F A286424 = (A005867(n) - A048863(n)) - 2*A117929(A002110(n))
%F A286424 = A048862(n) - 2*A117929(A002110(n)).
%e A286424 a(0) = 0 by definition. A002110(0) = 1; 1 is coprime to all numbers; the only possible totative pair is (1,1) and this does not include both a prime and a nonprime.
%e A286424 a(1) = 0 since, of the floor(A005867(1)/2) = 1 totative pair (1,1) of A002110(1) = 2, none include a both a prime and a nonprime.
%e A286424 a(2) = 1 since, the only totative pair (1,5) of A002110(1) = 6 includes both a prime and a nonprime.
%e A286424 a(3) = 1 since only (1,29) includes both a prime and a nonprime.
%e A286424 a(4) = 4 since (23,187), (41,169), (67,143), (89,121) include a both a prime and a nonprime.
%t A286424 Table[Function[P, Count[Prime@ Range[n + 1, PrimePi[P]], q_ /; ! PrimeQ[P - q]]]@ Product[Prime@ i, {i, n}], {n, 0, 9}] (* _Michael De Vlieger_, May 08 2017 *)
%Y A286424 Cf. A000849, A002110, A005867, A048862, A048863, A057704, A057705, A116979, A117929.
%K A286424 nonn,hard,more
%O A286424 0,5
%A A286424 _Michael De Vlieger_, May 08 2017
%E A286424 a(11) from _Giovanni Resta_, May 09 2017