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.

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, 62, 64, 69, 70, 72, 78, 80, 86, 88, 92, 94, 96, 100, 102, 108, 110, 115, 116, 118, 124, 126, 132, 134, 138, 140, 146, 148, 154, 156, 161, 162, 164, 170, 172, 178, 180

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..61.

Wikipedia, Fractional part

Wikipedia, Inequality of arithmetic and geometric means

Example

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.
4 is not a term because the geometric mean is 2.90..., whereas the arithmetic mean is 3.02..., and 0.90 > 0.02.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A356142/A102928 (the arithmetic mean of s(n)).

Sequence in context: A140199 A072190 A363658 * A177445 A022826 A053035

Adjacent sequences: A356134 A356135 A356136 * A356138 A356139 A356140

Keyword

nonn

Author

Mike Jones, Jul 27 2022

Extensions

More terms from Stefano Spezia, Jul 27 2022

Status

approved