OFFSET
1,1
COMMENTS
The smallest numbers k having n prime divisors are 128, 162, 3600, 558144, ... .
It is possible to find numbers where this property can be repeated, for example: 31250000 -> (31250000, 125000) -> (125000, 2500) -> (2500, 50) and these numbers have the same prime divisors 2 and 5.
More generally, it is possible to find pairs of numbers (x, y) of the form x = x(0)Ax(q) and y = y(0)By(q), x < y, with the same prime divisors such that B = p*A where A and B are the numbers representing the middle digits of x and y, and p is a prime divisor of x and y. Example: (a(67), a(70)) = (1263168, 1789488) have the same prime divisors {2,3,17,43} and 78948 = 3*26316.
EXAMPLE
558144 is in the sequence because the prime divisors of 558144 and 5814 are 2, 3, 17 and 19.
MAPLE
with(numtheory):
for n from 100 to 10000000 do:
x:=convert(n, base, 10):n1:=nops(x):
s:=sum('x[i]*10^(i-2) ', 'i'=2..n1-1):
if factorset(n) = factorset(s)
then
printf(`%d, `, n):
else
fi:
od:
MATHEMATICA
sdpdQ[n_]:=Module[{c=FromDigits[Rest[Most[IntegerDigits[ n]]]]}, FactorInteger[ n][[All, 1]]==FactorInteger[c][[All, 1]]]; Select[ Range[10, 125000], sdpdQ] (* Harvey P. Dale, Jan 30 2021 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Michel Lagneau, Jun 11 2014
STATUS
approved