A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

1, 2, 4, 6, 10, 16, 26, 42, 72, 124, 218, 390, 706, 1292, 2388, 4436, 8292, 15578, 29376, 55592, 105532, 200858, 383220, 732756, 1403848, 2694404, 5179938, 9973430, 19229826, 37125562, 71762396, 138871260, 269021848, 521666984, 1012520400, 1966957692, 3824240848

Offset: 1

Author

John W. Layman, Dec 05 2001

Keywords

• nonn

Comments

Also the number of subsets of {1,2,...,n} with sum of entries divisible by the largest element (compare A000016). See the Palmer Melbane link for a bijection. - Joel B. Lewis, Nov 13 2014

Examples

			a(4)=6, since {1}, {2}, {3}, {4}, {1,2,3} and {2,3,4} contain their averages.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(6) = 16 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                     {2,3,4}  {1,2,3}      {6}
                              {1,3,5}      {1,2,3}
                              {2,3,4}      {1,3,5}
                              {3,4,5}      {2,3,4}
                              {1,2,3,4,5}  {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {1,2,3,4,5}
                                           {2,3,4,5,6}
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3333
• Palmer Melbane, Combinations under constraint, Art of Problem Solving thread. - _Joel B. Lewis_, Nov 13 2014

Crossrefs

Subsets containing n whose mean is an element are A000016.

The version for integer partitions is A237984.

Subsets not containing their mean are A327471.

Cf. A051293, A063776, A066571, A135342, A324736, A325705, A326083, A326836, A327478, A327481.

Programs

Mathematica

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] &]; Apply[ Plus, 2^(i/a) * EulerPhi[a]]/i, {i, n}]/2, {n, 34}]
(* second program *)
Table[Length[Select[Subsets[Range[n]],MemberQ[#,Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

PARI

a(n) = (1/2)*sum(i=1, n, (1/i)*sumdiv(i, d, if (d%2, 2^(i/d)*eulerphi(d)))); \\ Michel Marcus, Dec 20 2020

Python

from sympy import totient, divisors
def A065795(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1 # Chai Wah Wu, Feb 22 2023

Formula

a(n) = (1/2)*Sum_{i=1..n} (f(i) - 1) where f(i) = (1/i) * Sum_{d | i and d is odd} 2^(i/d) * phi(d).

a(n) = (n + A051293(n))/2.

a(n) = 2^n - A327471(n). - Gus Wiseman, Sep 14 2019

Extensions

Edited and extended by Robert G. Wilson v, Nov 15 2002