A366642 Lexicographically earliest sequence of distinct primes such that the sequence of ratios (number of divisors of n that are coprime to these primes)/(number of divisors of n) has an asymptotic mean 1/2.

2, 3, 5, 149, 10771, 16575407, 39516855101743, 60095055821549024117399447, 96668175211190122501174866643973679330023904660323

Offset: 1

Author

Amiram Eldar, Oct 15 2023

Keywords

• nonn

Comments

The sequence of the number of divisors of n that are coprime to these primes is A366643.

Equivalently, a(n) is the lexicographically earliest sequence of distinct primes such that Product_{n>=1} (a(n)-1) * log(a(n)/(a(n)-1)) = 1/2.

The next term has 99 digits and is too large to be included in the data section.

Examples

			The asymptotic mean of (number of divisors of n that are coprime to 2)/A000005(n) = A001227(n)/A000005(n) is log(2) = 0.693... > 1/2. Therefore a(1) = 2.
The asymptotic mean of (number of divisors of n that are coprime to 2 and 3)/A000005(n) = A035218(n)/A000005(n) is 2*log(3/2)*log(2) = 0.562... > 1/2. Therefore a(2) = 3.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3 and 5)/A000005(n) is 8*log(5/4)*log(3/2)*log(2) = 0.501... > 1/2. Therefore a(3) = 5.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3, 5 and 7)/A000005(n) is 48*log(7/6)*log(5/4)*log(3/2)*log(2) = 0.464... < 1/2. Therefore a(4) is not 7.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3, 5 and 149)/A000005(n) is 1184*log(149/148)*log(5/4)*log(3/2)*log(2) = 0.50002... > 1/2, and 149 is the least prime with this property. Therefore a(4) = 149.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..13

Crossrefs

Cf. A000005, A001227, A035218, A366643.

Programs

Mathematica

g[x_] := -(x-1)*Log[1-1/x]; seq[len_] := Module[{s = {}, r = 1/2, p = 1}, Do[p = NextPrime[InverseFunction[g][r]]; AppendTo[s, p]; r /= g[p], {len}]; s]; seq[7]