A359930 - OEIS

%I #10 Jan 29 2023 10:45:31

%S 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,

%T 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,

%U 5,6,5,5,4,4,5,5,5,5,5,4,4,4,5,5,5,6

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

%C 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.

%H Hal M. Switkay, <a href="/A359930/b359930.txt">Table of n, a(n) for n = 1..9991</a>

%e 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.

%Y Cf. A005153, A182134.

%K nonn

%O 1,6

%A _Hal M. Switkay_, Jan 18 2023