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 A294907 #6 Nov 28 2017 15:50:10 %S A294907 87,925,4757,17699,43357,97703,187813,350321,595871,920081,1405609, %T A294907 2024047,2827861,3931217,5348053,7053941,9058607,11637667,14631209, %U A294907 18251339,22657429,27786589,33829567,40651799,48209237,56928409,67107197,78713287,92233283,107643667 %N A294907 a(n) is the smallest number k such that exactly half of the prime(n+1)-rough numbers in the interval [prime(n)^2 + 1, k] are prime. %C A294907 Students who are first learning about prime and composite numbers and factorization learn that trial division is a simple way to determine whether a given number N is prime, and that only those divisors up through the square root of N need to be tried, since trial division by each prime up through a given prime p is sufficient to completely factor every composite number up through at least p^2. %C A294907 Given a four-function calculator and a set of randomly-selected integers of manageable size to either factor or identify as prime, one can observe fairly quickly that as trial division is attempted using the primes in increasing order as divisors, many numbers that aren't divisible by any of the first several primes turn out to be prime. %C A294907 Given some positive integer j > prime(n)^2, if we consider the set of integers that exceed prime(n)^2 but do not exceed j, and we exclude each number that is divisible by any of the first n primes, then the numbers that remain will include exactly as many primes as composites if j = a(n), but if j < a(n), then most of the numbers that remain will be prime. %e A294907 The 1st prime is 2, and exactly half of the 42 3-rough numbers (i.e., odd numbers) in the interval [2^2 + 1, 87] are prime, and more than half of the 3-rough numbers in [5, k] are prime for all k < 87, so a(1)=87. %e A294907 The 2nd prime is 3, and exactly half of the 306 5-rough numbers (i.e., numbers that are not divisible by 2 or 3) in the interval [3^2 + 1, 925] are prime, and more than half of the 5-rough numbers in [10, k] are prime for all k < 925, so a(2) = 925. %Y A294907 Cf. A005408 (3-rough numbers, i.e., the odd numbers), A007310 (5-rough numbers), A007775 (7-rough numbers), A008364 (11-rough numbers). %K A294907 nonn %O A294907 1,1 %A A294907 _Jon E. Schoenfield_, Nov 10 2017