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Problems & Puzzles:
Puzzles
Puzzle 1284 The Right-Crescendo Serpent
On September 27, 2026, my
friend G. L. Honaker, Jr., sent the following
puzzle.
But first, here are some historical
notes:
a) The core ideas in this puzzle goes
back to 2005, when Amarnath Murthy published the
sequence
https://oeis.org/A110764
b) More recently G.
L. Honaker, Jr. in 2016 published this
Curio, corresponding to a 75 digits prime
obtained as a result of this sequence.
c) Michael
Branicky extended this sequence of digits from 71 to
85 in 2021
Anyhow, here goes the
puzzle:
Starting with any
chosen digit (1-9) as N, "Concatenate
the first
n
digits of this
number N
where the next digit equals the number of distinct
prime
factors
the number contains".
This is the
same than N = N*10+QDPF(N), QDPF = "Quantity of
Distinct Prime Factors"
By definition the
next digit can be only an integer between 0 to 9.
For example: if
the chosen first digit k is "1", then first 75
digits-number of the "Serpent-1" is this one :
102332234332543254432554756323145363256462444734575376546347484633443327927...
because:
1 -> 0
10 -> 2
102 -> 3
and so on...
Incidentally, the 75-digit number
shown above is a prime number. The same occurred when
the Serpent-1 was 102332234332543254432554756323. For
the purpose of this puzzle there is nothing special
because the Serpent becomes a prime number. It
simply means that the following digit is "1".
According to the definition of the growth of
this kind of numbers, its growth will end if the
factorization of N reaches to a QDPF>9.
Carlos Rivera has found that this Serpent-1 cannot
be larger than this number:
1023322343325432544325547563231453632564624447345753765463474846334433279271327473467566674355
(94 digits), because this number has 10 distinct
prime factors.
Here are the Largest Serpent-k
for K=1,2,4,5,6,7,9.
Starting
digit, k
|
Largest Right-k-Serpent, and digits |
|
1 |
102332234332543254432554756323145363256462444734575376546347484633443327927132
7473467566674355 (94) |
|
2 |
212223423223325645533283465474656566653445654682734728566554643545974785546466
(78) |
|
4 |
4112232431234254334546655672647444654665446544755546676632695466746667
(70) |
|
5 |
512124333323255345465454846545546646732356465434673355768557794745684
(69) |
|
6 |
62224245424554444944345313435553447955354344468565483384647455545375334575844
(77) |
|
7 |
7112322532431245442445436314543269536658746334456534784675758835664687358555
(76) |
|
9 |
9123223264544454533223454453455854654256444665
(46) |
It is reasonable to expect that something
similar will occur with k=3 & 8.
Q.
Can you get the maximum value for 3-Serpent and
8-Serpent?
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