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
Keywords
Examples
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.
Links
- Wikipedia, Fractional part
- Wikipedia, Inequality of arithmetic and geometric means
Programs
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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 *) -
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
Extensions
More terms from Stefano Spezia, Jul 27 2022
Comments