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 A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n. 18

%I

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

%C 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 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%H Antti Karttunen, <a href="/A316314/b316314.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>

%F a(n) = A316398(n) - 1.

%e The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.

%e 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 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]],{n,100}]

%o (PARI)

%o up_to = 65537;

%o A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }

%o v056239 = vector(up_to,n,A056239(n));

%o 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 Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299701, A299702, A301899, A301957, A304793, A316313.

%Y One less than A316398.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jun 29 2018

%E More terms from _Antti Karttunen_, Sep 23 2018

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Last modified April 23 13:25 EDT 2019. Contains 322386 sequences. (Running on oeis4.)