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A305103 Heinz numbers of connected integer partitions with z-density -1. 0
2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 171 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

First differs from A305078 at a(61) = 171, A305078(61) = 169.

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

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

195 is the Heinz number of {2,3,6} with corresponding multiset multisystem {{1},{2},{1,2}}, which is connected with z-density -1.

MATHEMATICA

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

zens[n_]:=If[n==1, 0, Total@Cases[FactorInteger[n], {p_, k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]]]];

Select[Range[300], And[zens[#]==-1, Length[zsm[primeMS[#]]]==1]&]

CROSSREFS

Cf. A001221, A056239, A112798, A286518, A302242, A303837, A304714, A304716, A305052, A305078, A305079.

Sequence in context: A229511 A305078 A328336 * A065520 A316428 A277702

Adjacent sequences:  A305100 A305101 A305102 * A305104 A305105 A305106

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 25 2018

STATUS

approved

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Last modified February 26 05:51 EST 2020. Contains 332277 sequences. (Running on oeis4.)