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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.
0
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
0,1
EXAMPLE
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.
PROG
(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 = ", ")
CROSSREFS
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Mar 29 2023
STATUS
approved