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

1,2

Comments

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

Links

Table of n, a(n) for n=1..24.

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

Example

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

Prog

(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

Crossrefs

Cf. A092670.

Sequence in context: A289671 A096853 A027155 * A129335 A264635 A046237

Adjacent sequences: A379332 A379333 A379334 * A379336 A379337 A379338

Keyword

nonn,more

Author

Ilya Gutkovskiy, Dec 21 2024

Extensions

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

Status

approved