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A051293 Number of nonempty subsets of {1,2,3,...,n} whose elements have an integer average. 35
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

Alois P. Heinz, Table of n, a(n) for n = 1..3332 (first 300 terms from T. D. Noe)

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.

MAPLE

with(numtheory):

b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:

a:= proc(n) option remember; `if`(n<1, 0, b(n)-1+a(n-1)) end:

seq(a(n), n=1..40);  # Alois P. Heinz, Jul 15 2019

MATHEMATICA

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

Table[Count[Subsets[Range[n]], _?(IntegerQ[Mean[#]]&)], {n, 35}] (* Harvey P. Dale, Apr 14 2018 *)

PROG

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

CROSSREFS

Row sums of A061865.

Cf. A114976.

Sequence in context: A154327 A074027 A018156 * A081660 A285291 A065618

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 August 24 18:12 EDT 2019. Contains 326295 sequences. (Running on oeis4.)