A357414 Number of nonempty subsets of {1..n} whose elements have an even geometric mean.

0, 0, 1, 1, 4, 4, 5, 5, 8, 12, 13, 13, 20, 20, 21, 21, 30, 30, 59, 59, 62, 62, 63, 63, 94, 104, 105, 187, 190, 190, 191, 191, 306, 306, 307, 307, 564, 564, 565, 565, 582, 582, 583, 583, 586, 600, 601, 601, 1120, 1134, 1275, 1275, 1278, 1278, 2125, 2125, 2144, 2144, 2145, 2145, 2360, 2360, 2361, 2381, 3938, 3938, 3939, 3939, 3942, 3942, 3943, 3943, 6560, 6560, 6561, 9663, 9666

Offset: 0

Author

Ilya Gutkovskiy, Sep 27 2022

Keywords

• nonn

Examples

			a(8) = 8 subsets: {2}, {4}, {6}, {8}, {1, 4}, {2, 8}, {1, 2, 4} and {2, 4, 8}.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..255

Crossrefs

Cf. A326027, A357356, A357412, A357413, A357416.

Programs

Python

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1 == 0
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1)
a = lambda n: b(n, 1, 0)
print([a(n) for n in range(26)]) # Michael S. Branicky, Sep 29 2022

Formula

a(p) = a(p-1) for prime p > 2. - Michael S. Branicky, Sep 30 2022

a(n) = A326027(n) - A357413(n). - Max Alekseyev, Mar 06 2025

Extensions

a(24)-a(41) from Michael S. Branicky, Sep 30 2022

Terms a(42) onward from Max Alekseyev, Oct 11 2023

a(0) prepended by Max Alekseyev, Mar 06 2025