A294080 Same-tree Moebius function of the multiorder of integer partitions indexed by Heinz numbers.

0, 1, 1, -1, 1, 0, 1, -1, -1, 0, 1, 2, 1, 0, 0, -2, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, -1, 0, 0, 0, 3, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, -3, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, -1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -2, 0, 1, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 8

Offset: 1

Author

Gus Wiseman, Feb 07 2018

Keywords

• sign

Comments

By convention a(1) = 0.

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

Crossrefs

Cf. A000041, A056239, A063834, A196545, A273873, A281145, A289501, A294018, A294019, A296150, A299202, A299203.

Programs

Mathematica

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
rmu[y_]:=rmu[y]=If[Length[y]===1,1,-Sum[Times@@rmu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,SameQ@@Total/@#]&]}]];
rmu/@ptns

PARI

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
muifbalancedfactorization(v) = if(!#v, 1, my(pw=A056239(v[1]), m=-1); for(i=1,#v,if(A056239(v[i])!=pw,return(0), m *= A294080(v[i]))); (m));
A294080aux(n, m, facs) = if(1==n, muifbalancedfactorization(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A294080aux(n/d, m, newfacs))); (s));
A294080(n) = if(1==n,0,if(isprime(n),1,A294080aux(n, n-1, List([]))));
\\ A memoized implementation:
map294080 = Map();
A294080(n) = if(1==n,0,if(isprime(n),1,if(mapisdefined(map294080,n), mapget(map294080,n), my(v=A294080aux(n, n-1, List([]))); mapput(map294080,n,v); (v)))); \\ Antti Karttunen, Sep 22 2018

Formula

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all same-trees (A281145, A294019) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.