A073608 a(1) = 1, a(n) = smallest number such that a(n) - a(n-k) is a prime power > 1 for all k.

1, 3, 5, 8, 10, 12

Offset: 1

Author

Amarnath Murthy, Aug 04 2002

Keywords

• nonn
• fini
• full

Comments

Differences |a(i)-a(j)| are prime powers for all i,j. Conjecture: sequence is bounded.

Proof that sequence is complete: Assume there is some k after the term 12. Then {k-1, k-3, k-5} must contain a multiple of 3. Also {k-8,k-10,k-12} also contains a multiple of 3. No prime > 12 is a multiple of 3, so the multiples of 3 are both prime powers. This implies there must be two powers of 3 that have a difference at most 11, but no such pair exists > 12 (only 1,3 and 3,9 qualify.) - Jim Nastos, Aug 09 2002

There is an elementary proof that no set of seven integers of this kind exists. - Don Reble, Aug 10 2002

Examples

			a(5) = 10 as 10-8, 10-5, 10-3, 10-1 or 2, 5, 7, 9 are prime powers.
		

Crossrefs

Cf. A073607.

Extensions

Sixth term from Jim Nastos, Aug 09 2002