A351243 Counterexamples to a conjecture of Selfridge and Lacampagne.

247, 277, 967, 977, 1211, 1219, 1895, 1937, 1951, 1961, 2183, 2191, 2911, 2921, 3029, 3641, 3649

Offset

1,1

Comments

The conjecture was that every natural number k not divisible by 3 can be written as the quotient of two terms chosen from A147991.

For every specific k, the problem of representing k as the quotient of two terms of A147991 can be decided by using a queue-based breadth-first search algorithm on the transition diagram of a finite automaton that on input j in base 3 computes j*k and checks to see if both j and j*k are in A147991.

It is not known if there are infinitely many counterexamples to the conjecture, but perhaps 3^m+4, for m >= 5 and odd, are.

References

R. K. Guy, Unsolved Problems in Number Theory, Springer, 2004. In Section F31, the conjecture of Selfridge and Lacampagne is mentioned, and it is stated that Don Coppersmith found the counterexample 247.

Links

Table of n, a(n) for n=1..17.

James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.

J. H. Loxton and A. J. van der Poorten, An Awful Problem About Integers in Base Four, Acta Arithmetica, volume 49, 1987, pages 193-203. In section 7, Selfridge and Lacampagne ask whether every k != 0 (mod 3) is the quotient of two terms of this sequence.

Crossrefs

Cf. A147991.

Sequence in context: A208188 A177212 A044983 * A175043 A051977 A177213

Adjacent sequences: A351240 A351241 A351242 * A351244 A351245 A351246

Keyword

nonn,base,more

Author

Jeffrey Shallit, Feb 05 2022

Status

approved