Problems & Puzzles: Puzzles Puzzle 107. K
Consecutive Smith numbers A
Smith number is such that the sum of
its digits is the sum of the digits of all its prime factors Example:
666 = 2*3*3*37 & 6+6+6 = 2+3+3+(3+7 But, are there consecutive Smithnumbers? In which extent? I have calculated the least Ktuplets of consecutive Smith numbers for K=1, 2, 3, 4 & 5.
Questions: 1.
Maybe
you would like to extend
the sequence Solution Shyam Sunder Gupta wrote the 7/2/2001: a) Consecutive Smiths mentioned for K= 4 & K=5 is incorrect , if primes are not considered as Smiths. Because for K= 4 , the numbers 2561678 ,2561679 & 2561680 are Smith numbers but 2561681 is prime. Similarly for K= 5 the numbers 4463535,4463536,4463537 & 4463538 are Smith numbers but 4463539 is prime. b) If primes are considered as Smiths then for K=1, the smith number 4 is incorrect. because 2,3,5 are primes so smiths and hence 2,3,4,5 are four consecutive Smith numbers. I feel that the purpose is to find consecutive nonprime Smith numbers. I have done a Systematic study of all Smith numbers up to 10^8 after writing code in Fortran and the correct results are as follows.
In fact there are only two sets of 5 consecutive numbers starting from 15966114 (as mentioned above) & 75457380 up to 10^8 . A further study reveals that up to 4*10^8, there are no 6consecutive Smith numbers and there are only 6 , 5consecutive Smith numbers including those mentioned above. *** More on this issue in Puzzle 247 *** 





