A359930 The number of practical numbers strictly between A005153(n) and A005153(n)^(1+1/n).

0, 0, 1, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 2, 2, 1, 3, 4, 3, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 3, 2, 3, 4, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3, 2, 3, 3, 6, 6, 5, 5, 5, 4, 4, 4, 6, 5, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6

Offset: 1

Author

Hal M. Switkay, Jan 18 2023

Keywords

• nonn

Comments

This sequence is based on Sun's conjecture that A005153(n)^(1/n) is a strictly decreasing sequence for n >= 3. This conjecture states for practical numbers what Firoozbakht's conjecture says for prime numbers, and implies that a(n) > 0 for n >= 3. It is valid at least for n <= 9991. The corresponding sequence for primes is A182134.

Examples

			The tenth practical number, A005153(10), is 24. 24^(1+1/10) is approximately 32.98. There are 3 practical numbers between 24 and 32.98, namely 28, 30, and 32. Thus a(10) = 3.
		

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..9991

Crossrefs

Cf. A005153, A182134.