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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.
6
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
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.
Sequence in context: A337862 A082295 A343293 * A272190 A060671 A269499
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 17 2019
STATUS
approved