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# User:Michael Beight

```Current interest:  Searching for numbers of the form:
.
F  =  P x (N) x Q    {the product of two primes, P <= Q, and N >= 1}
.
=  P + ... + Q    {also equal to the *sum* of all the primes from P to Q}
```

When N is 1, I call this a "strong Fuji" number (*). These are probably the rarest.

```  Examples:   10  =  2 x  5  =  2 +   3 +  5
39  =  3 x 13  =  3 + ... + 13
```

When N `fits' between P and Q (i.e., its smallest prime factor is >= P), I call it a "weak Fuji" number. This is A055233 (+)

```  Example:  2 935 561 623 745  =  5 x (19x53x61) x 9 557 887
```

When N does not fit between P and Q, I call it a "fauxji". This is A055514 (^)

```  Example:  10 225 245 560  =  503 x (2x2x2x5) x 508 213
```

All known Fujis and fauxjis:

```                           10  =  2 x (1) x 5
39  =  3 x (1) x 13
155  =  5 x (1) x 31
371  =  7 x (1) x 53
10 225 245 560  =  503 x (2x2x2x5) x 508 213
2 935 561 623 745  =  5 x (19x53x61) x 9 557 887
454 539 357 304 421  =  3 536 123 x (1) x 128 541 727
7 228 559 051 256 366 318  =  73 x (2x3x11x82067) x 18 281 691 653
1 390 718 713 078 158 117 206  =  370 794 889 x (2x7) x 267 902 967 061
..And no more < 1e22 .
```

Footnotes:

(*) The name `Fuji' came from a professor's webpage I came across in the late 90's who was working in Tokyo, Japan, and was using this term for these numbers (only when N fits between P and Q). Actually he called them "Mt. Fuji" numbers.

`Fauxji' came from the suggestion of a friend who likes to speak French. "Faux pas" means "false step". Fauxji is pronounced "FOE-jee".

(+) A055233 is all strong and weak Fujis.

(^) A055514 is all strong and weak Fujis and all fauxjis.

Keynumbers: 10225245560, 2935561623745, 454539357304421, 7228559051256366318, 1390718713078158117206