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A147752 Size of the largest subset of {1,2,3,...,n} whose geometric mean is an integer. 3
1, 1, 1, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n-1) <= a(n) <= max(a(n-1), nu_{A006530(n)}(n!)) where nu_p(n!) is the exponent of the largest power of p that divides n!. - Robert Israel, Jan 02 2018

Let k be the geometric mean of the subset. Then k is in A055932. - David A. Corneth, Jan 03 2018

LINKS

Table of n, a(n) for n=1..74.

David A. Corneth, Possible subsets of {1,2,3,...,n} giving a(n)

EXAMPLE

For n=4, (1*4)^(1/2)=2 and (1*2*4)^(1/3)=2. No other subset of {1,2,3,4} has integer geometric mean, so a(4)=3.

MAPLE

ub:= proc(k, n) local p, i, v, t;

  p:= max(numtheory:-factorset(k));

  t:= 0;

  for i from 1 do

    v:= floor(n/p^i);

    if v = 0 then return t fi;

    t:= t+v;

  od

end proc:

f:= proc(n) option remember; local goodk, m, u, s, S;

  m:= f(n-1);

  u:= ub(n, n);

  if u <= m then return m fi;

  goodk:= {1} union select(t -> ub(t, n) > m, {$2..n-1});

  S:= combinat:-subsets(goodk);

  while not S[finished] do

    s:= S[nextvalue]() union {n};

    if nops(s) <= m then next fi;

    if type(simplify(convert(s, `*`)^(1/nops(s))), integer) then m:= nops(s); if m = u then return m fi fi;

  od:

  m

end proc:

f(1):= 1:

seq(f(n), n=1..74); # Robert Israel, Jan 02 2018

MATHEMATICA

Array[Length@ Last@ Select[Subsets@ Range@ #, IntegerQ@ GeometricMean@ # &] &, 20] (* Michael De Vlieger, Jan 02 2018 *)

CROSSREFS

Cf. A006530, A147751, A147753.

Sequence in context: A227727 A050499 A304431 * A236682 A114227 A187469

Adjacent sequences:  A147749 A147750 A147751 * A147753 A147754 A147755

KEYWORD

nonn

AUTHOR

John W. Layman, Nov 11 2008

EXTENSIONS

a(1)-a(3) corrected and a(21)-a(74) from Robert Israel, Jan 02 2018

STATUS

approved

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Last modified October 23 09:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)