A361915 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.

13, 113, 1069, 5051, 18553, 44417, 99439, 190921, 356351, 603149, 933073, 1416223, 2044201, 2856559, 3957883, 5379287, 7093217, 9113263, 11693687, 14701529, 18345209, 22758829, 27879563, 33938257, 40808759, 48364003, 57099061, 67292237, 78919781, 92417891

Offset: 0

Author

Ya-Ping Lu, Mar 29 2023

Keywords

• nonn

Examples

			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.
   m:   2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
   N_p: 1, 2, 2, 3, 3, 4, 4, 4,  4,  5,  5,  6,  6,  6,  6,  7, ...
   N_c: 0, 0, 1, 1, 2, 2, 3, 4,  5,  5,  6,  6,  7,  8,  9,  9, ...
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).
   m:   3, 5, 7, ..., 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, ...
   N_p: 1, 2, 3, ..., 23, 23, 24, 24,  25,  26,  26,  27,  28,  28,  29,  29, ...
   N_c: 0, 0, 0, ..., 23, 24, 24, 25,  25,  25,  26,  26,  26,  27,  27,  28, ...
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.
		

Crossrefs

Cf. A000040, A002808, A072731, A097454.

Programs

Python

from sympy import isprime, prime
R = []; L = [x for x in range(2, 100000001)]
for n in range(30):
    np = 0; nc = 0; found = 0
    if n > 0: q = prime(n); L = [x for x in L if x%q != 0]
    for m in L:
        if isprime(m): np += 1; p = m
        else: nc += 1
        if np == nc: Lp = p; found = 1
    if found: R.append(Lp)
print(*R, sep = ", ")