A285573 - OEIS

A285573

Number of finite nonempty sets of pairwise indivisible divisors of n.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 19, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 19, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 49, 2, 5, 9, 7, 5, 19, 2, 9, 5, 19, 2, 34, 2, 5, 9, 9, 5, 19, 2, 20, 5, 5, 2, 49, 5, 5, 5, 14, 2, 49, 5, 9, 5, 5, 5, 27, 2, 9, 9, 19

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OFFSET

1,2

COMMENTS

From Robert Israel, Apr 21 2017: (Start)

If n = p^k for prime p, a(n) = k+1.

If n = p^j*q^k for distinct primes p,q, a(n) = binomial(j+k+2,j+1)-1. (End)

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

The a(12)=9 sets are: {1}, {2}, {3}, {4}, {6}, {12}, {2,3}, {3,4}, {4,6}.

MAPLE

g:= proc(S) local x, Sx; option remember;

if nops(S) = 0 then return {{}} fi;

x:= S[1];

Sx:= subsop(1=NULL, S);

procname(Sx) union map(t -> t union {x}, procname(remove(s -> s mod x = 0 or x mod s = 0, Sx)))

end proc:

f:= proc(n) local F, D;

F:= ifactors(n)[2];

D:= numtheory:-divisors(mul(ithprime(i)^F[i, 2], i=1..nops(F)));

nops(g(D)) - 1;

end proc:

map(f, [$1..100]); # Robert Israel, Apr 21 2017

MATHEMATICA

nn=50;

stableSets[u_, Q_]:=If[Length[u]===0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r===w||Q[r, w]||Q[w, r]], Q]]]];

Table[Length[Rest[stableSets[Divisors[n], Divisible]]], {n, 1, nn}]

CROSSREFS

Cf. A006126, A048143, A076078, A076413, A198085, A285572.

Sequence in context: A343654 A100565 A244098 * A325339 A010846 A073023

Adjacent sequences: A285570 A285571 A285572 * A285574 A285575 A285576

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 21 2017

STATUS

approved