A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 9, 4, 4, 4, 6, 2, 8, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 8, 2, 6, 6

Offset

1,2

Comments

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

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..16384

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

Formula

a(n) = A316314(n) + 1.

Example

The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Mean/@Subsets[primeMS[n]]]], {n, 100}]

Prog

(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A316398(n) = { my(m=Map(), s, k=0); fordiv(n, d, if((d>1)&&!mapisdefined(m, s = A056239(d)/bigomega(d)), mapput(m, s, s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

Crossrefs

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

One more than A316314.

Sequence in context: A369310 A318465 A073180 * A183095 A304817 A242802

Adjacent sequences: A316395 A316396 A316397 * A316399 A316400 A316401

Keyword

nonn

Author

Gus Wiseman, Jul 01 2018

Extensions

More terms from Antti Karttunen, Sep 23 2018

Status

approved