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

%I A194356 #43 Feb 19 2024 14:18:22
%S A194356 1,2,5,10,4,20,25,50,100,8,40,125,200,250,500,1000,16,80,400,625,1250,
%T A194356 2000,2500,5000,10000,32,160,800,3125,4000,6250,12500,20000,25000,
%U A194356 50000,100000,64,320,1600,8000,15625,31250,40000,62500,125000,200000,250000
%N A194356 Triangle of divisors of 10^n, each number occurring once.
%C A194356 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
%H A194356 T. D. Noe, <a href="/A194356/b194356.txt">Rows for n = 0..100</a>
%e A194356 The n-th row of the triangle begins with 2^n and ends with 10^n:
%e A194356    1;
%e A194356    2,  5,  10;
%e A194356    4, 20,  25,  50,  100;
%e A194356    8, 40, 125, 200,  250,  500, 1000;
%e A194356   16, 80, 400, 625, 1250, 2000, 2500, 5000, 10000;
%p A194356 T:={{1}}:
%p A194356 for n from 1 to 9 do
%p A194356   T:={op(T),numtheory[divisors](10^n) minus numtheory[divisors](10^(n-1))};
%p A194356 od:
%p A194356 T; # _Martin Renner_, Jan 16 2023
%t A194356 Join[{{1}}, Table[Complement[Divisors[10^n], Divisors[10^(n-1)]], {n, 9}]]
%o A194356 (Python)
%o A194356 from sympy import divisors
%o A194356 A194356 = []
%o A194356 for k in range(0,7): # shows the terms in the range 10^0 ... 10^6
%o A194356     for divisor in divisors(10**k):
%o A194356         if divisor not in A194356: A194356.append(divisor)
%o A194356 print(A194356) # _Karl-Heinz Hofmann_, Feb 19 2024
%o A194356 (Python)
%o A194356 from math import isqrt
%o A194356 def A194356(n):
%o A194356     exp = isqrt(n)
%o A194356     aarray = [2**exp, 10**exp]
%o A194356     while aarray[-1] % 2 == 0: aarray.append(aarray[-1]//2)
%o A194356     while aarray[0] * 5 < 10**exp: aarray = [aarray[0]*5] + aarray
%o A194356     return sorted(aarray)[n-exp**2]
%o A194356 print([A194356(n) for n in range(0,49)]) # _Karl-Heinz Hofmann_, Feb 19 2024
%o A194356 (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
%Y A194356 Cf. A000079 (1st column), A011557 (right diagonal).
%Y A194356 Cf. A003592 (numbers of the form 2^i*5^j).
%Y A194356 Cf. A194357 (divisors of 6^n), A194358 (divisors of 30^n).
%K A194356 nonn,tabf
%O A194356 0,2
%A A194356 _T. D. Noe_, Aug 25 2011