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 Quite a Smithonian! (Posted on 2010-08-21)
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 !

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 7-up | Comment 1 of 8

4 : 1st Smith number
4 = 2*2

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

73615 : 1st number of 1st 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 : 1st number of 1st 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 : 1st number of 1st 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 : 1st number of 1st 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 : 1st number of 1st 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

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