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

1,6

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

Links

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

Example

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.

Mathematica

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

Prog

(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

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.

Sequence in context: A008647 A036475 A330746 * A337531 A316556 A187279

Adjacent sequences: A316552 A316553 A316554 * A316556 A316557 A316558

Keyword

nonn

Author

Gus Wiseman, Jul 06 2018

Extensions

More terms from Antti Karttunen, Sep 28 2018

Status

approved