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A243812 Let x(0)x(1)... x(q-1)x(q) denote the decimal expansion of a number n. The sequence lists the numbers such that n and the number represented by its middle digits x(1)x(2)...x(q-1) have the same distinct prime divisors. 0
128, 162, 768, 864, 1024, 1568, 1728, 2048, 2500, 3200, 3600, 3773, 3993, 5184, 6400, 8464, 9600, 10368, 12500, 14884, 20577, 25000, 29768, 32000, 32400, 36000, 37500, 59264, 62500, 64000, 64800, 67500, 83544, 86784, 96000, 97200, 100842, 118528, 125000 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..39.

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:

CROSSREFS

Sequence in context: A257446 A130445 A114565 * A194766 A257211 A109651

Adjacent sequences:  A243809 A243810 A243811 * A243813 A243814 A243815

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, Jun 11 2014

STATUS

approved

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Last modified September 25 00:48 EDT 2020. Contains 337333 sequences. (Running on oeis4.)