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A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

%I A327475 #13 Jan 13 2024 17:05:26
%S A327475 1,2,3,6,9,16,27,46,77,136,239,426,769,1400,2571,4762,8857,16568,
%T A327475 31139,58734,111165,211044,401695,766418,1465489,2807672,5388783,
%U A327475 10359850,19946833,38459624,74251095,143524762,277742489,538043664,1043333935,2025040766,3933915349
%N A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.
%F A327475 a(n) = A051293(n) + 1.
%e A327475 The a(0) = 1 through a(5) = 16 subsets:
%e A327475   {}  {}   {}   {}       {}       {}
%e A327475       {1}  {1}  {1}      {1}      {1}
%e A327475            {2}  {2}      {2}      {2}
%e A327475                 {3}      {3}      {3}
%e A327475                 {1,3}    {4}      {4}
%e A327475                 {1,2,3}  {1,3}    {5}
%e A327475                          {2,4}    {1,3}
%e A327475                          {1,2,3}  {1,5}
%e A327475                          {2,3,4}  {2,4}
%e A327475                                   {3,5}
%e A327475                                   {1,2,3}
%e A327475                                   {1,3,5}
%e A327475                                   {2,3,4}
%e A327475                                   {3,4,5}
%e A327475                                   {1,2,4,5}
%e A327475                                   {1,2,3,4,5}
%p A327475 with(numtheory):
%p A327475 b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:
%p A327475 a:= proc(n) option remember; `if`(n=0, 1, b(n)-1+a(n-1)) end:
%p A327475 seq(a(n), n=0..36);  # _Alois P. Heinz_, Jan 13 2024
%t A327475 Table[Length[Select[Subsets[Range[n]],#=={}||IntegerQ[Mean[#]]&]],{n,0,10}]
%o A327475 (Python)
%o A327475 from sympy import totient, divisors
%o A327475 def A327475(n): return 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))-n+1 # _Chai Wah Wu_, Feb 22 2023
%Y A327475 If the subset is required to contain n, we get A063776.
%Y A327475 Cf. A000016, A051293, A065795, A082550, A135342, A327474, A327481.
%K A327475 nonn
%O A327475 0,2
%A A327475 _Gus Wiseman_, Sep 13 2019