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%I #11 Jun 16 2023 13:43:39
%S 13,113,1069,5051,18553,44417,99439,190921,356351,603149,933073,
%T 1416223,2044201,2856559,3957883,5379287,7093217,9113263,11693687,
%U 14701529,18345209,22758829,27879563,33938257,40808759,48364003,57099061,67292237,78919781,92417891
%N a(n) is the smallest prime p such that, for m >= nextprime(p), there are more composites than primes in the range [2, m], where multiples of primes prime(1) through prime(n) are excluded.
%e The number of primes, N_p, and the number of composite, N_c, in the range [2, m] are listed in the table below, where N_p = N_c occurs at m = 9, 11 and 13. For m >= nextprime(13) = 17, N_c > N_p. So, a(0) = 13 is the case for n = 0, in which none of the multiples of primes is excluded from the integer list.
%e m: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
%e N_p: 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, ...
%e N_c: 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 8, 9, 9, ...
%e If the multiples of prime(1) are excluded from the list, 113 is the smallest prime such that N_c > N_p for m >= nextprime(113) = 127 and, thus, a(1) = 113 (see below).
%e m: 3, 5, 7, ..., 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, ...
%e N_p: 1, 2, 3, ..., 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, ...
%e N_c: 0, 0, 0, ..., 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, ...
%e If multiples of prime(1) and prime(2) are excluded, a(2) = 1069. If multiples of prime(1), prime(2) and prime(3) are excluded, a(3) = 5051.
%o (Python)
%o from sympy import isprime, prime
%o R = []; L = [x for x in range(2, 100000001)]
%o for n in range(30):
%o np = 0; nc = 0; found = 0
%o if n > 0: q = prime(n); L = [x for x in L if x%q != 0]
%o for m in L:
%o if isprime(m): np += 1; p = m
%o else: nc += 1
%o if np == nc: Lp = p; found = 1
%o if found: R.append(Lp)
%o print(*R, sep = ", ")
%Y Cf. A000040, A002808, A072731, A097454.
%K nonn
%O 0,1
%A _Ya-Ping Lu_, Mar 29 2023