Problems & Puzzles: Puzzles

Puzzle 247.  Consecutive Smith numbers

The largest sequence of consecutive Smith numbers was gotten by Shyam Sunder Gupta approximately on February 2001 as an answer to a not very elegant puzzle of these pages (Puzzle 107).

These days the largest sequence found by Shyam was composed by 5 members (4, 728, 73615, 4463535, 15966114) published as A059754, and thus remained the same until two days ago when I tried to cover a small range over the scanned by Shyam (<4x10^8).

Unexpectedly the sixth member of this sequence came up very fast (24 hours later) in my not very powerful PC: 2050918644.

So I think that with the new speedy machines we have nowadays we should get one or two more terms for this sequence.

Q1      Find more terms to A059754

As you know prime numbers are trivial Smith numbers. What if include primes as Smith numbers discarding the compositeness from its definition?. I have gotten the following sequence (defining each member as the earliest number of a sequence of k consecutive Smith numbers, primes included as Smith numbers)

2, 2, 2, 2, 1458855, 1790478, 429990136, ?

Q2      Find more terms to the last sequence

Solution:

Faride Firoozbakht reports:

"Next term of the sequence A059754 (4,728,73615,4463535,15966114, 2050918644)is greater than 2442000000 and next term of the sequence 2,2,2,2,1458855,1790478,429990136 is greater than 701000000."

***

Shyam wrote (July 2007):

Q1.Next term of the sequence A059754 (4,728,73615,4463535,15966114, 2050918644)is greater than 10^10. Refer My webpage on "Smith Numbers" http://www.shyamsundergupta.com/smith.htm

Q2.Next term of the sequence 2,2,2,2,1458855,1790478,429990136 is 4475873320"

***

J. K. Andersen wrote (April 08):

Q1) The seventh term of A059754 is 164736913905; the first start of 7 consecutive Smith numbers.

***

Giovanni Resta wrote on January 2014:

I have a further term for Puzzle 247, question Q2. It is 1979414080360, the 9-th term.

By the way, that sequence was missing from OEIS, so I submitted it.
Now it is https://oeis.org/draft/A235812 (I do not know when it will be accepted because in these days there is quite a long backlog at OEIS).

***

 Records   |  Conjectures  |  Problems  |  Puzzles