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A114976 Number of subsets of {1,2,....,n} with an arithmetic mean that is an integer and also a divisor of n.

%I A114976 #20 Apr 25 2023 13:21:59
%S A114976 1,2,2,5,2,14,2,30,11,80,2,280,2,764,128,2557,2,9036,2,29656,1958,
%T A114976 103134,2,373454,119,1300824,36992,4681568,2,17119030,2,61799636,
%U A114976 758982,226451040,2180,837469677,2,3084255132,16391220,11451833394,2,42746493556,2
%N A114976 Number of subsets of {1,2,....,n} with an arithmetic mean that is an integer and also a divisor of n.
%C A114976 a(n) <= A051293(n);
%C A114976 a(n) = 2 iff n is prime, just as for the number of divisors of n and also, at least for the very first terms, a(n)=odd iff n is a square: these observations might suggest conjectures on a deeper relationship with A000005.
%H A114976 Alois P. Heinz, <a href="/A114976/b114976.txt">Table of n, a(n) for n = 1..100</a>
%e A114976 a(9) = 11: {1}, {3}, {9}, {1,5}, {2,4}, {1,2,6}, {1,3,5}, {2,3,4}, {1,2,3,6}, {1,2,4,5} and {1,2,3,4,5}, e.g. also {1,4,7} has an integral arithmetic mean, but (1+4+7)/3 = 4 is not a divisor of 9.
%p A114976 b:= proc(n, m, s, c) option remember; `if`(n=0,
%p A114976       `if`(c>0 and denom(s)=1 and irem(m, s)=0, 1, 0),
%p A114976        b(n-1, m, s, c)+b(n-1, m, (s*c+n)/(c+1), c+1))
%p A114976     end:
%p A114976 a:= proc(n) option remember; forget (b); b(n$2, 0$2) end:
%p A114976 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 15 2019
%t A114976 b[n_, m_, s_, c_] := b[n, m, s, c] = If[n==0, If[c>0 && Denominator[s]==1 && Mod[m, s]==0, 1, 0], b[n-1, m, s, c]+b[n-1, m, (s c + n)/(c+1), c+1]];
%t A114976 a[n_] := b[n, n, 0, 0];
%t A114976 Array[a, 50] (* _Jean-François Alcover_, Nov 04 2020, after _Alois P. Heinz_ *)
%Y A114976 Cf. A000005, A051293.
%K A114976 nonn
%O A114976 1,2
%A A114976 _Reinhard Zumkeller_, Feb 22 2006
%E A114976 a(27)-a(38) from _Donovan Johnson_, Jun 10 2010
%E A114976 a(39)-a(43) from _Alois P. Heinz_, Jul 15 2019