A327471 Number of subsets of {1..n} not containing their mean.

1, 1, 2, 4, 10, 22, 48, 102, 214, 440, 900, 1830, 3706, 7486, 15092, 30380, 61100, 122780, 246566, 494912, 992984, 1991620, 3993446, 8005388, 16044460, 32150584, 64414460, 129037790, 258462026, 517641086, 1036616262, 2075721252, 4156096036, 8320912744, 16658202200

Offset

0,3

Links

Table of n, a(n) for n=0..34.

Formula

a(n) = 2^n - A065795(n). - Alois P. Heinz, Sep 13 2019

Example

The a(1) = 1 through a(5) = 22 subsets:
  {}  {}     {}     {}         {}
      {1,2}  {1,2}  {1,2}      {1,2}
             {1,3}  {1,3}      {1,3}
             {2,3}  {1,4}      {1,4}
                    {2,3}      {1,5}
                    {2,4}      {2,3}
                    {3,4}      {2,4}
                    {1,2,4}    {2,5}
                    {1,3,4}    {3,4}
                    {1,2,3,4}  {3,5}
                               {4,5}
                               {1,2,4}
                               {1,2,5}
                               {1,3,4}
                               {1,4,5}
                               {2,3,5}
                               {2,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}

Mathematica

Table[Length[Select[Subsets[Range[n]], !MemberQ[#, Mean[#]]&]], {n, 0, 10}]

Prog

(Python)
from sympy import totient, divisors
def A327471(n): return (1<<n)-(sum((sum(totient(d)<<k//d-1 for d in divisors(k>>(~k&k-1).bit_length(), generator=True))<<1)//k for k in range(1, n+1))>>1) # Chai Wah Wu, Feb 22 2023

Crossrefs

Subsets containing their mean are A065795.

Subsets containing n but not their mean are A327477.

Partitions not containing their mean are A327472.

Strict partitions not containing their mean are A240851.

Cf. A000016, A007865, A051293, A067538, A082550, A114639, A135342, A324741, A327476.

Sequence in context: A181158 A018108 A033497 * A239075 A260916 A192627

Adjacent sequences: A327468 A327469 A327470 * A327472 A327473 A327474

Keyword

nonn

Author

Gus Wiseman, Sep 12 2019

Extensions

More terms from Alois P. Heinz, Sep 13 2019

Status

approved