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A356137 Positive integers m such that the fractional part of the geometric mean of the sequence s(m) does not exceed the fractional part of the arithmetic mean of s(m), where s(m) is the sequence 1 + 1/1, 2 + 1/2, ..., m + 1/m.

%I A356137 #60 Jul 31 2022 23:11:03
%S A356137 1,2,3,5,6,8,10,13,14,16,18,22,24,26,30,32,34,38,40,42,46,48,54,56,61,
%T A356137 62,64,69,70,72,78,80,86,88,92,94,96,100,102,108,110,115,116,118,124,
%U A356137 126,132,134,138,140,146,148,154,156,161,162,164,170,172,178,180
%N A356137 Positive integers m such that the fractional part of the geometric mean of the sequence s(m) does not exceed the fractional part of the arithmetic mean of s(m), where s(m) is the sequence 1 + 1/1, 2 + 1/2, ..., m + 1/m.
%C A356137 The idea is to take note of when the fractional parts of the geometric mean and arithmetic mean "follow suit" with respect to the celebrated geometric mean <= arithmetic mean inequality.
%H A356137 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fractional_part">Fractional part</a>
%H A356137 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means">Inequality of arithmetic and geometric means</a>
%e A356137 2 is a term because the geometric mean of 1 + 1/1 and 2 + 1/2 is the geometric mean of 2 and 2.5, which is a bit less than 2.24, whereas the arithmetic mean of 2 and 2.5 is 2.25, and 0.24 <= 0.25.
%e A356137 4 is not a term because the geometric mean is 2.90..., whereas the arithmetic mean is 3.02..., and 0.90 > 0.02.
%t A356137 max=180; a={}; s[m_]:=m+1/m; For[m=1,m<=max,m++,If[FractionalPart[Mean[Table[s[k],{k,m}]]] >= FractionalPart[GeometricMean[Table[s[k],{k,m}]]],AppendTo[a,m]]]; a (* _Stefano Spezia_, Jul 27 2022 *)
%o A356137 (PARI) isok(m) = my(v=vector(m, k, k+1/k)); frac(sqrtn(vecprod(v), m)) <= frac(vecsum(v)/m); \\ _Michel Marcus_, Jul 28 2022
%Y A356137 Cf. A356142/A102928 (the arithmetic mean of s(n)).
%K A356137 nonn
%O A356137 1,2
%A A356137 _Mike Jones_, Jul 27 2022
%E A356137 More terms from _Stefano Spezia_, Jul 27 2022