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A316314
Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.
18
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, 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, 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
OFFSET
1,6
COMMENTS
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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
a(n) = A316398(n) - 1.
EXAMPLE
The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.
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.
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]], {n, 100}]
PROG
(PARI)
up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
v056239 = vector(up_to, n, A056239(n));
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 29 2018
EXTENSIONS
More terms from Antti Karttunen, Sep 23 2018
STATUS
approved