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A111233 Number of nonempty subsets of {1, 1/2, 1/3, ..., 1/n} that sum to an integer. 2
1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 5, 5, 5, 11, 11, 11, 21, 21, 43, 43, 43, 43, 83, 83, 83, 83, 255, 255, 449, 449, 449, 895, 895, 1407, 2111, 2111, 2111, 2111, 4159, 4159, 8319, 8319 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

If the set was {1/2, 1/3, 1/4, ..., 1/n}, that is, the set is lacking the element 1, then the sequence would be (a(n)-1)/2. - Robert G. Wilson v, Sep 23 2006

LINKS

Table of n, a(n) for n=1..43.

FORMULA

a(p^e) = a(p^e-1). - Robert G. Wilson v, Sep 23 2006

EXAMPLE

1, 1/2 + 1/3 + 1/6 = 1 and 1 + 1/2 + 1/3 + 1/6 = 2 are integers, so a(6)=3.

MATHEMATICA

Needs["DiscreteMath`Combinatorica`"]

f[1] = 1; f[n_] := Block[{c = 0, k = 2, lmt = 2^n/2, int = Range[2, n]}, While[k < lmt, If[IntegerQ[Plus @@ (1/NthSubset[k, int])], c++ ]; k++ ]; 2c+1];

Do[Print[{n, f[n] // Timing}], {n, 40}]

(* Robert G. Wilson v, Sep 23 2006 *)

(* Second program (not needing Combinatorica): *)

a[n_] := a[n] = If[n == 1, 1, If[PrimePowerQ[n], a[n-1], Count[Total /@ Subsets[1/Range[n], {1, 2^(n-1)}], _?IntegerQ]]];

Table[Print[n, " ", a[n] // Timing]; a[n], {n, 1, 25}] (* Jean-François Alcover, Aug 11 2022 *)

PROG

(Python)

from fractions import Fraction

from functools import lru_cache

@lru_cache(maxsize=None)

def b(n, soh, c):

if n == 0: return int(soh.denominator == 1)

return b(n-1, soh, c) + b(n-1, soh+Fraction(1, n), c+1)

a = lambda n: b(n, 0, 0) - 1 # subtract empty set

print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Aug 11 2022

CROSSREFS

Sequence in context: A333819 A216944 A178832 * A210746 A283986 A343515

Adjacent sequences: A111230 A111231 A111232 * A111234 A111235 A111236

KEYWORD

nonn,hard,more

AUTHOR

John W. Layman, Oct 28 2005

EXTENSIONS

More terms from Robert G. Wilson v, Sep 23 2006

STATUS

approved

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Last modified December 9 11:21 EST 2022. Contains 358700 sequences. (Running on oeis4.)