cp's OEIS Frontend

A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

%I A316314 #11 Sep 23 2018 21:26:15
%S A316314 0,1,1,1,1,3,1,1,1,3,1,4,1,3,3,1,1,4,1,4,3,3,1,5,1,3,1,4,1,5,1,1,3,3,
%T A316314 3,5,1,3,3,5,1,7,1,4,4,3,1,6,1,4,3,4,1,5,3,5,3,3,1,8,1,3,4,1,3,7,1,4,
%U A316314 3,7,1,7,1,3,4,4,3,7,1,6,1,3,1,8,3,3,3,5,1,7,3,4,3,3,3,7,1,4,4,5,1,7,1,5,5
%N A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.
%C A316314 A rational number q is a nonempty-subset-average of an integer partition y if there exists a nonempty submultiset of y with average q.
%C A316314 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A316314 Antti Karttunen, <a href="/A316314/b316314.txt">Table of n, a(n) for n = 1..65537</a>
%H A316314 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A316314 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%F A316314 a(n) = A316398(n) - 1.
%e A316314 The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.
%e A316314 The a(72) = 7 subset-averages of (2,2,1,1,1) are 1, 5/4, 4/3, 7/5, 3/2, 5/3, 2.
%t A316314 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A316314 Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]],{n,100}]
%o A316314 (PARI)
%o A316314 up_to = 65537;
%o A316314 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
%o A316314 v056239 = vector(up_to,n,A056239(n));
%o A316314 A316314(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = v056239[d]/bigomega(d)), mapput(m,s,s); k++)); (k); }; \\ _Antti Karttunen_, Sep 23 2018
%Y A316314 Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299701, A299702, A301899, A301957, A304793, A316313.
%Y A316314 One less than A316398.
%K A316314 nonn
%O A316314 1,6
%A A316314 _Gus Wiseman_, Jun 29 2018
%E A316314 More terms from _Antti Karttunen_, Sep 23 2018