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A072207
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a(0) = 1; for n>0, a(n) = number of distinct sums of subsets of {1, 1/2, 1/3, 1/4, ..., 1/n} (allowing the empty subset).
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5
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1, 2, 4, 8, 16, 32, 52, 104, 208, 416, 832, 1664, 1856, 3712, 7424, 9664, 19328, 38656, 59264, 118528, 126976, 224128, 448256, 896512, 936832, 1873664, 3747328, 7494656, 7771136, 15542272, 15886336, 31772672, 63545344, 112064512, 224129024, 231010304, 237031424, 474062848, 948125696
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OFFSET
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0,2
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COMMENTS
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Define L to be a set of rational numbers with L={0}, s=1 in generation 0 and the rule "replace each term t in L with terms t-1/s, t+1/s, then increment s" to generate the next generation. a(n) is the size of the set in generation n. First generation = {-1,1}, second generation = {-3/2,-1/2,1/2,3/2}, 3rd generation = {-11/6,-7/6,-5/6,-1/6,1/6,5/6,7/6,11/6}. - Dylan Hamilton, Oct 28 2010
If n is a prime power, a(n) = 2*a(n-1). However, this is not "if and only if", e.g., a(10) = 2*a(9). - Robert Israel, Nov 23 2016
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LINKS
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FORMULA
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a(p) = 2 * a(p-1) for p prime. a(2*p) = 2 * a(2*p-1) for p>3 prime. - Giovanni Resta, Jul 18 2019
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MAPLE
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S[1]:= {0, 1}: A[1]:= 2:
for n from 2 to 30 do
S[n]:= S[n-1] union (1/n +~ S[n-1]);
A[n]:= nops(S[n]);
od:
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MATHEMATICA
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w = {0}; o = {1}; s = 1
Do[w = Union[Flatten[{w - (1/s), w + (1/s)}]]; AppendTo[o, Length[w]]; ++s, {NumberOfApplications}]; o # Dylan Hamilton, Oct 28 2010
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PROG
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(PARI) M72207=List([[0]]); A072207(n)={for(k=#M72207, n, listput(M72207, setunion(Set([x+1/k|x<-M72207[k]]), M72207[k]))); #M72207[n+1]} \\ M. F. Hasler, Oct 29 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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