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[ October 3, 2000 ]
Sets of three composites in bi-directional
'Sum Of Prime Factors' progression/retrogression

I'd like to submit to the devoted number lovers the following rather hard puzzle
( catalogued as Sloane's A057874 ) in the hope of receiving the elusive  4 th
set of three composites in bi-directional
'Sum Of Prime Factors' progression/retrogression.
A mouthful, I am aware!
Hence this jolly table which will clarify the concept better than any words can.

Many more triplets have been found [ October, 2000 ]
by  Roberto Botrugno  from Italy... read on!

Click here to see how Roberto
managed to find these remarkably large solutions

95174191298687992
 + 5.19 =
down		up
– 7.17 =		 + 373.467 =
down		up
– 383.457 =		 + 2.2.2.3.179.251.277 =
down		up
– 2.2.17.53.179.463 =		 
119175031298688708
 + 7.17 =
down		up
– 11.13 =		 + 383.457 =
down		up
– 397.443 =		 + 2.2.17.53.179.463 =
down		up
– 2.2.2.2.11.19.179.499 =		 
143175871298689424
! 1  SOPF = 24! 2  SOPF = 840! 3  SOPF = 716  right 179 is a common factor!!
checkIn the first set you'll notice no doubt six consecutive primes 5, 7, 11, 13, 17 and 19.
checkAll the members of the first two sets are semi-primes i.e. having exactly two distinct factors.
checkThe third set has a common prime divisor > 2 to all members namely 179.
checkCurious is also the loop taken counterclockwise of the middle digits
of the factors of the second set starting from 443.
It starts with a 4, then a 5, then a ... up to and then finally a 9 in factor 397.
373 * 467
383 * 457
397 * 443

Common prime divisor of the members of the composite triplet
(298687992, 298688708, 298689424) which are in a bi-directional
'sum of prime factors' (i.e., 716) progression/retrogression.

Some OEIS entries
A050780 - (n + sopf_n = m) and (m - sopf_m = n). Sequence gives values of n.
A050781 - (n + sopf_n = m) and (m - sopf_m = n). Sequence gives values of m.
A057874 - Sets of three composites in bidirectional 'sum of prime factors' progression/retrogression.


A000078 Prime Curios! Prime Puzzle
Wikipedia 78 Le Nombre 78 Numberland 78

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