A327474 Number of distinct means of subsets of {1..n}, where {} has mean 0.

1, 2, 4, 6, 10, 16, 26, 38, 56, 78, 106, 138, 180, 226, 284, 348, 420, 500, 596, 698, 818, 946, 1086, 1236, 1408, 1588, 1788, 2000, 2230, 2472, 2742, 3020, 3328, 3652, 3996, 4356, 4740, 5136, 5568, 6018, 6492, 6982, 7512, 8054, 8638, 9242, 9870, 10520, 11216

Offset: 0

Author

Gus Wiseman, Sep 13 2019

Keywords

• nonn

Examples

			The a(3) = 6 distinct means are 0, 1, 3/2, 2, 5/2, 3.
		

Crossrefs

The version for only nonempty subsets is A135342.

Cf. A000016, A051293, A065795, A082550, A327475, A327481.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 2, 4, 6][n+1],
      2*a(n-1)-a(n-2)+numtheory[phi](n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 22 2023

Mathematica

Table[Length[Union[Mean/@Subsets[Range[n]]]],{n,0,10}]

Python

from itertools import count, islice
from sympy import totient
def A327474_gen(): # generator of terms
    a, b = 4, 6
    yield from (1,2,4,6)
    for n in count(3):
        a, b = b, (b<<1)-a+totient(n)
        yield b
A327474_list = list(islice(A327474_gen(),30)) # Chai Wah Wu, Feb 22 2023

Formula

a(n) = A135342(n) + 1.

a(n) = 2*a(n-1) - a(n-2) + phi(n-1) for n>3. - Chai Wah Wu, Feb 22 2023