

A309366


When the positive integers are written as products of primes in nondecreasing order, a(n) is the least prime to occur more frequently in nth position than in any other position.


2




OFFSET

1,1


COMMENTS

In such products of primes, prime(m) occurs in nth 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 nth position than (n1)th position.
Primes p satisfying a(n) <= p < a(n+1) appear to occur more frequently in nth position than in any other position.
The next term, a(5), is estimated to be ~ 6*10^11.


LINKS

Table of n, a(n) for n=1..4.


FORMULA

a(1) = prime(1) = 2.
For n >= 2, a(n) = min{ k : k = prime(m), A281890(m,n) > A002110(m) * A281890(m,n1) }.


EXAMPLE

a(1) = prime(1) = 2, since 2 occurs in nth 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

Cf. A002110, A027746, A126283, A281889, A281890.
Sequence in context: A167218 A013045 A290864 * A007506 A042693 A328746
Adjacent sequences: A309363 A309364 A309365 * A309367 A309368 A309369


KEYWORD

nonn,more


AUTHOR

Peter Munn, Jul 25 2019


STATUS

approved



