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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.
1
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.
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
KEYWORD
nonn
AUTHOR
Mike Jones, Jul 27 2022
EXTENSIONS
More terms from Stefano Spezia, Jul 27 2022
STATUS
approved