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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 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.)