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.

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

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:

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

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