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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