User:Michael Beight

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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}

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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 .

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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.

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