A Smith number is a composite number for which, the sum of its digits is equal to the sum of the digits in its prime factorization.
By definition, the factors are treated as digits.
For example, 22 =2*11 is a Smith numberbecause 2 + 2 = 2 + 1 + 1.
Another example: 666 = 2*3*3*37 and 6+6+6 = 2+3+3+(3+7 ).

The following table provides the least number of the 1st occurrence of K consecutive Smith numbersfor K=1, 2, 3, …

Please extend it as far as you are able to.

K ......First member of the least K-tuplet
1 ......4
2 ......728
3 ......73615**
4 .......FIND OUT
5 .......FIND OUT
6 +++ How far can you reach?


**Rem: The third number is 73615, since (73615, 73616, 73617) is the first example of 3 consecutive Smith numbers.
G o o d L u c k !

4 : 1^st Smith number
  4 = 2*2

728 : 1^st number of 1^st Smith Brothers
  7+2+8 = 2*3+7+(1+3)
  7+2+9 = 3*6

73615 : 1^st number of 1^st Smith Triples
  7+3+6+1+5 = 5+(1+4+7+2+3)
  7+3+6+1+6 = 2*4+(4+3)+(1+0+7)
  7+3+6+1+7 = 3+(5+3)+(4+6+3)

4463535 : 1^st number of 1^st Smith Quadruples
  4+4+6+3+5+3+5 = 3+5+(2+9)+(3+1)+(3+3+1)
  4+4+6+3+5+3+6 = 2*4+7+(1+1)+(3+6+2+3)
  4+4+6+3+5+3+7 = (1+3)+(1+7)+(1+9)+(1+0+6+3)
  4+4+6+3+5+3+8 = 2+3+(7+4+3+9+2+3)

15966114 : 1^st number of 1^st Smith Quintuples
  1+5+9+6+6+1+1+4 = 2+3+(7+1+9)+(3+7+0+1)
  1+5+9+6+6+1+1+5 = 5+(1+1)+(4+3)*2+(1+5+7)
  1+5+9+6+6+1+1+6 = 2*2+(3+1)+(3+3+1)+(3+8+9)
  1+5+9+6+6+1+1+7 = 3*2+(2+3)+(1+3+7)+(5+6+3)
  1+5+9+6+6+1+1+8 = 2+7+(1+9)+(1+9+3)+(3+1+1)

2050918644 : 1^st number of 1^st Smith Sextuples
  2+0+5+0+9+1+8+6+4+4 
       = 2*2+3+(3+3+7)+(5+0+7+1+5+1)
  2+0+5+0+9+1+8+6+4+5 
       = 5+(4+1+0+1+8+3+7+2+9)
  2+0+5+0+9+1+8+6+4+6
       = 2+7+(3+1)+(4+1)+(1+1+5+2+5+9)
  2+0+5+0+9+1+8+6+4+7
       = 3+(5+9)+(1+1+5+8+7+1+1+1)
  2+0+5+0+9+1+8+6+4+8 
       = 2*3+(2+3)+(7+3)+(1+0+7)+(1+4+2+7)
  2+0+5+0+9+1+8+6+4+9 
       = (1+3)+(4+7)+(1+6+3)+(2+0+5+9+3)

164736913905 : 1^st number of 1^st Smith Septuples
  1+6+4+7+3+6+9+1+3+9+0+5
       = 3*2+5+(2+5+7)+(1+4+2+4+4+4+3+7)
  1+6+4+7+3+6+9+1+3+9+0+6
       = 2+(6+1)+(1+0+9)+(1+3+9)+(8+9+1+2+3)
  1+6+4+7+3+6+9+1+3+9+0+7
       = (3+1)*2+(2+2+9+3)+(7+4+7+5+9)
  1+6+4+7+3+6+9+1+3+9+0+8
       = 2*2+3+7+(1+9+6+1+1+5+3+7+3+7)
  1+6+4+7+3+6+9+1+3+9+0+9
       = (3+9+4+7)+(4+1+7+3+7+2+4+7)
  1+6+4+7+3+6+9+1+3+9+1+0 
       = 2+5+(1+9)+(4+1)+(2+1+1+4+7+2+2+9)
  1+6+4+7+3+6+9+1+3+9+1+1
       = 3+(2+9+3)+(1+8+7+4+1+4+0+0+9)

Edited on August 24, 2010, 9:26 am

Posted by Dej Mar on 2010-08-21 14:55:48