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Number of prime factors of a natural number

If the number of prime factors of a natural number $n$ is $k$, how can we express $n$ in terms of $k$?
I tried using
$$n=\sum_{1\leq i\leq k}i^n,$$
but I got an expression $k^k$, where $n$ can take any value $n>k>0$.

A:

$n=\sum_{1\leq i\leq k}i^n$ corresponds to the fact that $i^n$ is divisible by the prime $i$. If we put $k-1$ back into the sum we get $n=k(k-1)+\sum_{1\leq i\leq k-1}i^n$.

1. Field of the Invention
The present invention relates to an image display device, such as an optical projector, and an operation method for the image display device.
2. Description of the Related Art
Optical projectors have been widely used as a projector for use in presentations at meetings and the like, and have been utilized in wide-range areas from domestic application to overseas application, such as in presentations for the purpose of education or training.
FIG. 4 is an illustration showing a schematic construction of a conventional image display device. In the image

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Pieces is the eighth studio album by American indie rock band The Pixies. It was released in 1997 by Elektra Records. In 1998, the album was voted the 63rd greatest album of all time by the American magazine Consequence of Sound. It was the band’s final studio album before the break-up of the band in 2001.

The first single from the album was “Bam Thwok”, which was released in February 1997. It was a very successful track, eventually reaching number 31 on the UK Singles Chart. Many of the songs on the album were written in the period after the previous album, Doolittle. “Self-Awareness” was also released as a single in the UK.

Critical reception

In his review for AllMusic, Stephen Thomas Erlewine describes Pieces as “the best album the Pixies made in the ’90s”, and it is ranked number 63 on the “Top 100 Albums of the 1990s” list by Rolling Stone magazine. In a retrospective review, Dave Thompson of musicOMH writes “This marks the band’s return to the sonic and thematic sound of their first three albums, but much of the music is now dressed up, highlighted by an extensive use of strings, keyboards and brass”.

Legacy
To date, the album is the only one from the Pixies era to not be released on a physical CD. After the band’s breakup, the group’s next official release was the live album Tour ’94-’95.

Track listing

Personnel

The Pixies
Kim Deal – vocals, guitars, bass guitar, percussion
Charles Thompson – guitars, keyboards, backing vocals
Joey Santiago – drums, percussion