A194356 Triangle of divisors of 10^n, each number occurring once.

1, 2, 5, 10, 4, 20, 25, 50, 100, 8, 40, 125, 200, 250, 500, 1000, 16, 80, 400, 625, 1250, 2000, 2500, 5000, 10000, 32, 160, 800, 3125, 4000, 6250, 12500, 20000, 25000, 50000, 100000, 64, 320, 1600, 8000, 15625, 31250, 40000, 62500, 125000, 200000, 250000

Offset: 0

Author

T. D. Noe, Aug 25 2011

Keywords

• nonn
• tabf

Comments

The following rule for divisibility applies: for each term t in the n-th row of the triangle, a positive integer m is divisible by t if the last n digits of m are divisible by t; e.g., for n = 2, since 20 is one of the terms in the 2nd row of the triangle, a number m is divisible by 20 if the last 2 digits of m are divisible by 20. - Martin Renner, Jan 15 2023

Examples

			The n-th row of the triangle begins with 2^n and ends with 10^n:
   1;
   2,  5,  10;
   4, 20,  25,  50,  100;
   8, 40, 125, 200,  250,  500, 1000;
  16, 80, 400, 625, 1250, 2000, 2500, 5000, 10000;
		

Links

• T. D. Noe, Rows for n = 0..100

Crossrefs

Cf. A000079 (1st column), A011557 (right diagonal).

Cf. A003592 (numbers of the form 2^i*5^j).

Cf. A194357 (divisors of 6^n), A194358 (divisors of 30^n).

Programs

Maple

T:={{1}}:
for n from 1 to 9 do
  T:={op(T),numtheory[divisors](10^n) minus numtheory[divisors](10^(n-1))};
od:
T; # Martin Renner, Jan 16 2023

Mathematica

Join[{{1}}, Table[Complement[Divisors[10^n], Divisors[10^(n-1)]], {n, 9}]]

PARI

row(n) = my(pow2 = 2^n, pow5 = 5^n); Set(concat(vector(n+1, i, pow5*2^(i-1)), vector(n,i,pow2*5^(i-1)))) \\ David A. Corneth, Feb 19 2024

Python

from sympy import divisors
A194356 = []
for k in range(0,7): # shows the terms in the range 10^0 ... 10^6
    for divisor in divisors(10**k):
        if divisor not in A194356: A194356.append(divisor)
print(A194356) # Karl-Heinz Hofmann, Feb 19 2024

Python

from math import isqrt
def A194356(n):
    exp = isqrt(n)
    aarray = [2**exp, 10**exp]
    while aarray[-1] % 2 == 0: aarray.append(aarray[-1]//2)
    while aarray[0] * 5 < 10**exp: aarray = [aarray[0]*5] + aarray
    return sorted(aarray)[n-exp**2]
print([A194356(n) for n in range(0,49)]) # Karl-Heinz Hofmann, Feb 19 2024