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A305831 Number of connected components of the strict integer partition with FDH number n. 3
0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

Let f = A050376. The FD-factorization of 1683 is 9*11*17 = f(6)*f(7)*f(10). The connected components of {6,7,10} are {{7},{6,10}}, so a(1683) = 2.

MATHEMATICA

FDfactor[n_]:=If[n===1, {}, Sort[Join@@Cases[FactorInteger[n], {p_, k_}:>Power[p, Cases[Position[IntegerDigits[k, 2]//Reverse, 1], {m_}->2^(m-1)]]]]];

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]]]]]]]]];

nn=200; FDprimeList=Array[FDfactor, nn, 1, Union]; FDrules=MapIndexed[(#1->#2[[1]])&, FDprimeList];

Table[Length[zsm[FDfactor[n]/.FDrules]], {n, nn}]

CROSSREFS

Cf. A048143, A050376, A064547, A213925, A299755, A299756, A304714, A304716, A305078, A305079, A305829, A305830, A305832.

Sequence in context: A196564 A196563 A198890 * A022927 A063435 A262352

Adjacent sequences:  A305828 A305829 A305830 * A305832 A305833 A305834

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 10 2018

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)