A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 19, 19, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 39, 39, 40, 40, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 62, 62, 63, 63, 64, 64, 65, 65

Offset

1,3

Comments

The geometric mean of a subset such as in name must be an odd number in {1..n} which might ease the search for terms. - David A. Corneth, Sep 29 2022

Links

Table of n, a(n) for n=1..70.

Formula

a(2*n-1) = a(2*n) for n >= 1. - David A. Corneth, Sep 29 2022

Example

a(9) = 7 subsets: {1}, {3}, {5}, {7}, {9}, {1, 9} and {1, 3, 9}.

Prog

(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
@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) if n&1 else b(n-1, p, c)
@lru_cache(maxsize=None)
def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 29 2022

Crossrefs

Cf. A001055, A326027, A357355, A357411, A357414, A357415.

Sequence in context: A035396 A326077 A005138 * A318296 A035679 A337587

Adjacent sequences: A357410 A357411 A357412 * A357414 A357415 A357416

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 27 2022

Extensions

a(24)-a(34) from Michael S. Branicky, Sep 29 2022

More terms from David A. Corneth, Sep 29 2022

Status

approved