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Search: id:A051293
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| A051293 |
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Number of subsets of {1,2,3,...,n} whose elements have an integer average. |
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6
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| 1, 2, 5, 8, 15, 26, 45, 76, 135, 238, 425, 768, 1399, 2570, 4761, 8856, 16567, 31138, 58733, 111164, 211043, 401694, 766417, 1465488, 2807671, 5388782, 10359849, 19946832, 38459623, 74251094, 143524761, 277742488, 538043663, 1043333934
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) is asymptotic to 2^(n+1)/n. More precisely, I conjecture for any m>0 : a(n)= {2^(n+1)/n} * {sum(k=0,m, A000670(k)/n^k) + o(1/n^(m+1))} (A000670 = preferential arrangements of n labeled elements) which can be written a(n) = {2^n/n} * {2 + sum(k=1,m, A000629(k)/n^k) + o(1/n^(m+1))} (A000629 = necklaces of sets of labeled beads). In fact I conjecture a(n)= {2^(n+1)/n} * {1+1/n+ 3/n^2+13/n^3+75/n^4+541/n^5+o(1/n^5)}. - Benoit Cloitre, Oct 20, 2002
A082550(n) = a(n+1) - a(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 19 2006
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REFERENCES
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63rd Annual William Lowell Putnam Mathematical Competition (Problem A3), Mathematics Magazine 76 (2003),76-80.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..300
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FORMULA
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a(n) = sum_{i=1..n} (A063776(i) - 1).
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EXAMPLE
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a(4)=8 because each of the 8 subsets {1}, {2}, {3}, {4}, {1, 3}, {2, 4}, {1, 2, 3}, {2, 3, 4} has an integer average.
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MATHEMATICA
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Table[ Sum[a = Select[Divisors[i], OddQ[ # ] & ]; Apply[Plus, 2^(i/a)*EulerPhi[a]]/i, {i, 1, n}] - n, {n, 1, 34}]
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PROGRAM
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(PARI) a(n)=sum(k=1, n, sumdiv(k, d, d%2*2^(k/d)*eulerphi(d))/k-1)
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CROSSREFS
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Cf. A114976.
Sequence in context: A154327 A074027 A018156 this_sequence A081660 A065618 A080084
Adjacent sequences: A051290 A051291 A051292 this_sequence A051294 A051295 A051296
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KEYWORD
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nonn,nice
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AUTHOR
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John W. Layman (layman(AT)math.vt.edu), Oct 30 1999
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EXTENSIONS
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Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 16 2002
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