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

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

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

Crossrefs

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

One less than A316398.

Sequence in context: A344852 A372720 A095960 * A183093 A183096 A029356

Adjacent sequences: A316311 A316312 A316313 * A316315 A316316 A316317

Keyword

nonn

Author

Gus Wiseman, Jun 29 2018

Extensions

More terms from Antti Karttunen, Sep 23 2018

Status

approved