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A350782 a(n) is the number of pairs of primes (p,q), p < q, such that (p + q)/2 = n!.

%I A350782 #18 Jan 25 2022 06:18:39
%S A350782 1,5,18,60,315,1615,9928,70437,637504,5829386,64647125,722750400
%N A350782 a(n) is the number of pairs of primes (p,q), p < q, such that (p + q)/2 = n!.
%C A350782 a(n) is the number of pairs of primes that are equidistant from n!.
%C A350782 Equivalently, a(n) is the number of positive integers d such that n! - d and n! + d are primes. For the smallest such d, iff there are no primes in the open interval (n! - d, n! + d), then n is a term in A053709, n! is a term in A053710, and d is a term in A053711.
%F A350782 a(n) = A002375(n!).
%e A350782 For n = 4, n! = 24, from which 5 pairs of primes are equidistant; in order of increasing distance, these are (19, 29), (17, 31), (11, 37), (7, 41), and (5, 43), so a(4) = 5. (A prime (23) lies between 19 and 29, so n=4 is not a term of A053709.)
%e A350782 For n = 5, n! = 120, from which 18 pairs of primes are equidistant: (113, 127), (109, 131), (103, 137), (101, 139), (89, 151), (83, 157), (73, 167), (67, 173), (61, 179), (59, 181), (47, 193), (43, 197), (41, 199), (29, 211), (17, 223), (13, 227), (11, 229), and (7, 233), so a(5) = 18. (The pair least distant from 120 is (113, 127), and there are no primes between 113 and 127, so n = 5 is a term of A053709, n! = 120 is a term of A053710, and d = 120 - 113 = 127 - 120 = 7 is a term of A053711.)
%t A350782 Table[Length@IntegerPartitions[2n!,2,Prime@Range@PrimePi[2n!]],{n,3,9}] (* _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%o A350782 (Python)
%o A350782 from sympy import isprime, nextprime, factorial
%o A350782 def A350782(n):
%o A350782     m, p, c = factorial(n), 3, 0
%o A350782     while p <= m:
%o A350782         if isprime(2*m-p):
%o A350782             c += 1
%o A350782         p = nextprime(p)
%o A350782     return c # _Chai Wah Wu_, Jan 16 2022
%Y A350782 Cf. A002375, A053709, A053710, A053711.
%K A350782 nonn,more
%O A350782 3,2
%A A350782 _Jon E. Schoenfield_, Jan 15 2022
%E A350782 a(14) from _Martin Ehrenstein_, Jan 25 2022