A335318 Harmonic numbers (A001599) k with a record abundancy index sigma(k)/k.

1, 6, 140, 270, 672, 8190, 30240, 332640, 14303520, 17428320, 27027000, 191711520, 2144862720, 3506025600, 5943057120, 14378364000, 45578332800, 288662774400, 505159855200, 2020639420800, 10680522652800, 54557264361600

Offset

1,2

Comments

The corresponding record values are 1, 2, 2.4, 2.666..., 3, 3.2, 4, 4.363..., ...

The terms 1, 6, 672 and 30240 are multiply perfect numbers (A007691) with abundancy indices 1, 2, 3, and 4, respectively. There is no 5-multiperfect number (A046060) in this sequence since A046060(1) = 14182439040 is larger than the harmonic number 5943057120 which is 5-abundant, having an abundancy index 5.067...

Links

Table of n, a(n) for n=1..22.

Example

The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their abundancy indices are 1, 2, 2, 2.4, 2.666..., 2 and 3. The record values, 1, 2, 2.4, 2.666... and 3 occur at 1, 6, 140, 270 and 672, the first 5 terms of this sequence.

Mathematica

rm = 0; s = {}; Do[h = DivisorSigma[0, n]/(r = DivisorSigma[1, n]/n); If[IntegerQ[h] && r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

Crossrefs

Cf. A000203, A001599, A335316, A335317.

Sequence in context: A288565 A090944 A007340 * A342358 A122483 A335369

Adjacent sequences: A335315 A335316 A335317 * A335319 A335320 A335321

Keyword

nonn,more

Author

Amiram Eldar, May 31 2020

Status

approved