A316557 Number of distinct integer averages of subsets of the integer partition with Heinz number n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 3, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 3, 2, 1, 3, 1, 3, 3, 1, 2, 4, 1, 4, 2, 4, 1, 2, 1, 2, 2, 2, 2, 5, 1, 3, 1, 3, 1, 4, 3, 2, 3, 4, 1, 3, 3, 3, 2, 3, 2, 2, 1, 3, 3, 3, 1, 4, 1, 2, 3

Offset: 1

Author

Gus Wiseman, Jul 06 2018

Keywords

• nonn

Comments

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

Examples

			The a(78) = 5 distinct integer averages of subsets of (6,2,1) are {1, 2, 3, 4, 6}.
		

Links

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

Crossrefs

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316555, A316556.

Programs

Mathematica

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

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));
A316557(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&(1==denominator(s = v056239[d]/bigomega(d)))&&!mapisdefined(m,s), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

Formula

a(n) <= A316314(n). - Antti Karttunen, Sep 25 2018

Extensions

More terms from Antti Karttunen, Sep 25 2018