A379335 Number of subsets of {-n..n} whose sum of reciprocals is 1.

1, 2, 4, 8, 16, 48, 96, 192, 384, 768, 1536, 5632, 11264, 22528, 77312, 154624, 309248, 922624, 1845248, 6848512, 17096576, 34193152, 68386304, 272849152

Offset: 1

Author

Ilya Gutkovskiy, Dec 21 2024

Keywords

• nonn
• more

Comments

Number of ways of writing 1 as Sum_{k=-n..n, k<>0} e(k)/k, where e(k) is 0 or 1.

Examples

			a(3) = 4 subsets: {1}, {-3, 1, 3}, {-2, 1, 2}, {-3, -2, 1, 2, 3}.
		

Crossrefs

Cf. A092670.

Programs

Python

from functools import cache
from fractions import Fraction
@cache
def b(i, s):
    if i == 0: return 1 if s == 1 else 0
    return b(i-1, s) + b(i-1, s+Fraction(1, (-1)**(i&1)*((i+1)>>1)))
a = lambda n: b(2*n, 0)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Dec 21 2024

Formula

a(n) <= 2*a(n-1) since we count s and s union {-1/n, 1/n} for each subset s counted in a(n-1); equality holds for n prime (and other cases). - Michael S. Branicky, Dec 21 2024

Extensions

a(12)-a(24) from Michael S. Branicky, Dec 21 2024