Problems & Puzzles: Puzzles Puzzle 210. Jeff's numbers Jeff Heleen sent the following nice puzzle:
I would like to add the following extensions to the Jeff's puzzle: Q.2 Find a titanic H(k) (disregarding the "earliest" condition) Q.3 Find the earliest Jeff's number H(2) expressible in two different ways as a product of two (not prime) factors Solution: Contributions came from J. C. Rosa, John Warf, J. K. Andersen & Johan Wiesenbauer. J. C. Rosa wrote for question 3:
I asked him to avoid solutions ending in zero and this is what he replied in return:
Later he added:
*** John Warf wrote for question 2:
*** J. K. Andersen wrote for questions 1, 2 & 3:
*** Johan Wiesenbauer wrote for question 1:
*** J. C. Rosa continued his search for the question 3. Here are summarized his results (4/4/03):
k  earliest H(k)
earliest H(k)
 divisible by
10 not divisible by 10

2  1260=6*210
105264=51*2064

=21*60 =204*516

3  13950=5*9*310
2669436=6*462*963

=5*31*90 =6*642*693

=9*31*50 =42*66*963

4  1169370=3*9*61*710
168478464=4*66*784*814

=3*9*71*610 =7*66*88*4144

=3*61*71*90 =7*66*448*814

=9*30*61*71 =7*88*444*616
Later he added:
Two news:
1°) The earliest H(5) expressible in
five different ways
divisible by 10 is
:137979450=5*7*9*93*4710
=5*7*9*471*930
=5*7*90*93*471
=5*9*70*93*471
=7*9*50*93*471
(About H(5) not divisible by 10 ,
I have found this number:
18738929664 which is expressible in 8
different ways
of 5 factors ! but maybe it's not the earliest )
*** One week ahead he added:
In order to complete my contribution to the
puzzle 210,
please , can you add the following result :
" The earliest H(5) not divisible by 10
is : 1949688832
1949688832=4*8*8*91*83692
=4*8*91*98*6832
=4*61*98*98*832
=8*8*49*91*6832
=8*49*61*98*832
*** J. C. Rosa continues obtaining new interesting results related to this issue. On May 2003, he wrote:
***





