A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

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

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).

Number of distinct values obtained when A289508 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 28 2018

Examples

			455 is the Heinz number of (6,4,3) which has possible GCDs of nonempty submultisets {1,2,3,4,6} so a(455) = 5.
		

Links

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

Crossrefs

Cf. A056239, A108917, A122768, A275972, A289508, A289509, A296150, A316313, A316314, A316430, A316556, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Programs

Mathematica

Table[Length[Union[GCD@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

PARI

A289508(n) = gcd(apply(p->primepi(p),factor(n)[,1]));
A316555(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A289508(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 28 2018

Extensions

More terms from Antti Karttunen, Sep 28 2018