A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2744, 2916, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400, 6724

Offset

1,1

Links

Table of n, a(n) for n=1..46.

Wikipedia, Geometric mean

Example

36 has two such factorizations: (3*12) and (4*9).
The sequence of terms together with their prime indices begins:
    36: {1,1,2,2}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   144: {1,1,1,1,2,2}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   324: {1,1,2,2,2,2}
   400: {1,1,1,1,3,3}
   484: {1,1,5,5}
   512: {1,1,1,1,1,1,1,1,1}
   576: {1,1,1,1,1,1,2,2}
   676: {1,1,6,6}
   784: {1,1,1,1,4,4}
   900: {1,1,2,2,3,3}
  1000: {1,1,1,3,3,3}
  1024: {1,1,1,1,1,1,1,1,1,1}
  1156: {1,1,7,7}
  1296: {1,1,1,1,2,2,2,2}
  1444: {1,1,8,8}

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[1000], Length[Select[facs[#], !IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]

Crossrefs

A subsequence of A001597.

Factorizations with integer mean are A326622.

Factorizations with integer geometric mean are A326028.

Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.

Sequence in context: A337862 A082295 A343293 * A272190 A060671 A269499

Adjacent sequences: A326663 A326664 A326665 * A326667 A326668 A326669

Keyword

nonn

Author

Gus Wiseman, Jul 17 2019

Status

approved