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By Cliff Pickower

From 1 to 100

Tau Numbers

Tau  numbers are those that  number of divisors of n divides n 
You can see the definition here A033950 and the list of the first 10,000 of these numbers here.
The current year, 2016, is one of these numbers because the number of divisors of 2016 is 36, and 2016/36 = 56.

If we subtract from 2016 his number of divisors 36, we obtain 1980, which is also a number Tau. 
We follow with the same procedure with the numbers obtained while Tau numbers are obtain.

2016 (36) -> 1980 (36) ---> 1944 (24) ---> 1920 (32) and here the chain ends because 1888 is not a number Tau.
 
We can say that 2016 generates (counting himself) four numbers Tau.


What is the longest string of numbers Tau that applying this methodology can be found?

Spanish Version : 1453 - Números Tau

Multiples with all the digits

Find for each prime, the sequence of multiples of this prime with the least sum, which contains each of the digits from 0 to 9, one and only one time.

Examples 

For 2 :  14, 36, 58, 70, 92. Sum = 270
For 3 :  12, 39, 48, 57, 60. Sum = 216
For 5 :  12345, 67890      . Sum = 80235
For 7 :  63, 70, 189, 245  . Sum = 567
For 11: 165, 704, 2398    . Sum = 3267


Can you check if this secuences are that with the least sum ?
Can you find the sequences for larger primes?

Update: Mmonchi and Vicente found this values

For 3: 6, 9, 18, 27, 30, 45 (135)
For 5: 13685, 24790 (38475) Mmonchi
For 7: 7, 28, 49, 63, 105 (252) Mmonchi
For 11: 264,539,1078 (1881)  Vicente
For 13: 26, 78, 195, 403 (702) Mmonchi
For 17: 34, 85, 102, 697 (918) Mmonchi
For 19: 19, 76, 285, 304 (684) Mmonchi
For 23: 46,92,713,805 (1656) Vicente
For 29: 493,580,1276 (2349) Vicente.
For 31: 372,496,1085 (1953) Vicente.

 
Spanish version : 1450 - Múltiplos con todos los dígitos

To split a number in primes in n ways

23 It is the smallest number that is the concatenation of two primes 2 and 3.   
237 is the smallest number that can be separated into two primes in two different ways: 2-37, 23-7  
2337 is the smallest number that can be separated into two primes ​​in three different ways: 2-337, 23-37, 233-7 
29397 is the smallest number that can be separated into two primes ​​in four different ways: 2-9397, 29-397, 293-97, 2939-7 


Hence we have the sequence 23, 237, 2337, 29397 
What is the next term in this sequence?


The smallest square that can be divided into two primes is 25 
  5²  = 25  and 25 can be divided into two primes 2-5 
77² = 5929  and this is the smallest square that  can be divided into two primes in two different ways: 59-29 and 5-929 
1549² = 2399401 and this is the smallest square that can be divided into two primes ​​in three different ways: 2399-401, 23-99401 and 2-399401.


23077² = 532547929 and this is the smallest square that can be divided into two primes ​​in four different ways: 5-32547929 ,  53-2547929 ,  532547-929 and  5325479-29

Hence we have the sequence 5, 77, 1549, 23077 


What is the next term in this sequence?