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A051293 Number of subsets of {1,2,3,...,n} whose elements have an integer average. 7
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; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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, Feb 19 2006

LINKS

T. D. Noe, Table of n, a(n) for n=1..300

63rd Annual William Lowell Putnam Mathematical Competition, Problem A3, Mathematics Magazine 76 (2003), 76-80.

FORMULA

a(n) = sum_{i=1..n} (A063776(i) - 1).

EXAMPLE

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.

MATHEMATICA

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] & ]; Apply[Plus, 2^(i/a)*EulerPhi[a]]/i, {i, 1, n}] - n, {n, 1, 34}]

PROG

(PARI) a(n)=sum(k=1, n, sumdiv(k, d, d%2*2^(k/d)*eulerphi(d))/k-1)

CROSSREFS

Cf. A114976.

Sequence in context: A154327 A074027 A018156 * A081660 A065618 A257548

Adjacent sequences:  A051290 A051291 A051292 * A051294 A051295 A051296

KEYWORD

nonn,nice

AUTHOR

John W. Layman, Oct 30 1999

EXTENSIONS

Extended by Robert G. Wilson v, Oct 16 2002

STATUS

approved

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Last modified December 8 12:55 EST 2016. Contains 278945 sequences.