A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

3, 37, 42719, 5737850066077

Offset: 1

Author

Peter Munn, Mar 26 2017

Keywords

• nonn
• more
• hard

Comments

The proportion of all integers that satisfy the divisibility criterion for p=prime(m) is determined using the proportion that satisfy it over any interval of primorial(m)=A002110(m) integers.

a(4) is from De Koninck, 2009; calculation credited to David Grégoire.

a(5) is about 7.887*10^34 assuming the Riemann Hypothesis, and about 7*10^34 unconditionally (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

Examples

			Exactly half of the integers are divisible by 2, so a(1)>2. Two-thirds of all integers are divisible by 2 or 3, so a(1) = 3.
		

References

• Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, pp. 13, 216 and 368.

Links

• Jean-Marie De Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
• Gérald Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, Erdős Centennial, Janos Bolyai Math. Soc., 2013, 651-681. HAL Id: hal-01281530.

Crossrefs

Cf. A002110, A077761, A096294, A194156, A281889.

Cf. A038110, A038111, A342479, A342480, A378720, A378721.

Formula

a(n) is least p=prime(m) such that 2*Sum_{k=0..n-1} A096294(m,k) < A002110(m).

log(log(a(n))) = n - b + O(1/sqrt(n)), where b = 1/3 + A077761 (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

Extensions

Definition edited by N. J. A. Sloane, Apr 01 2017