OFFSET
1,1
COMMENTS
In such products of primes, prime(m) occurs in n-th position A281890(m,n) times in every interval of A002110(m)^n positive integers, as explained in A281890. A002110(m) = primorial(m), product of first m primes.
For n >= 2, a(n) is the least prime to occur more frequently in n-th position than (n-1)-th position.
Primes p satisfying a(n) <= p < a(n+1) appear to occur more frequently in n-th position than in any other position.
The next term, a(5), is estimated to be ~ 6*10^11.
FORMULA
EXAMPLE
a(1) = prime(1) = 2, since 2 occurs in n-th position when an integer divisible by 2^n is written as a product of primes in nondecreasing order, thus more frequently in 1st position than in other positions.
Prime(2) = 3 occurs more often in 1st position than 2nd position, specifically once for every 6 consecutive integers (since A281890(2,1) = 1 and primorial(2) = 6) compared with 5 times for every 36 consecutive integers (since A281890(2,2) = 5 and primorial(2)^2 = 36). As 2 and 3 each occur more frequently in 1st position than 2nd position, a(2) > 3.
Prime(3) = 5 occurs in 1st position A281890(3,1) = 2 times in primorial(3) = 30, in 2nd position A281890(3,2) = 62 times in 30^2, in 3rd position A281890(3,3) = 1322 times in 30^3, and decreasingly frequently in subsequent positions. 2/30 < 62/30^2 and 62/30^2 > 1322/30^3. Thus 5 occurs most frequently in 2nd position and is the first prime to do so, so a(2) = 5.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Munn, Jul 25 2019
STATUS
approved