A308247 a(n) is the least integer not the difference of two prime(n)-smooth numbers.

5, 41, 281, 1849, 9007, 35803

Offset

1,1

Comments

The known terms have been found by exhaustive search and then proved not to be the difference of prime(n)-smooth numbers using assertions such as <a,b,...> +- a(n) !== <c,d,...> (modulo m) meaning that no element of the subgroup of Z/m generated by a,b,... added to a(n) is congruent modulo m with an element of the subgroup generated by <c,d,...>. For example: <2> +- 41 !== <3> (mod 91) and the fact that 41+1 is not 3-smooth is enough to prove that 41 is not the difference of 3-smooth numbers; <2> + 281 !== <3,5> (mod 13981), <2> - 281 !== <3,5> (mod 76627) and <3> +- 281 !== <2,5> along with the fact that 281+1 is not 5-smooth is enough to show that 281 is not the difference of 5-smooth numbers. The proofs get exponentially harder as n increases. For example, <2, 11> + 9007 !== <3, 5, 7> (mod 308859288230831), or <2,5,7> + 35803 !== <3,11,13> (mod 2219897250633559197203).

The next few terms are conjectured to be 158857, 681179, 2516509, 10772123, 51292187, 186323681; if they were not, they would provide examples of ABC-triples with quality greater than 2.

Links

Table of n, a(n) for n=1..6.

Esteban Crespi de Valldaura, Proof that a(n) is not prime(n)-smooth for n=2,3,4,5,6

Wikipedia, abc conjecture

Example

We see that 1 = 2-1, 2 = 4-2, 3 = 4-1, and 4 = 8-4. It is easy to see that 5 is not the difference of two powers of 2, so a(1) = 5. In the same way we can see that all the integers up to 40 are the difference of 3-smooth numbers, but as shown above 41 is not, so a(2)=41.

Crossrefs

P-smooth_numbers: A000079, A003586, A051037, A002473, A051038, ...

a(i) is the first term in each of A101082, A290365, A308456, A326318, A326319, A326320.

Sequence in context: A198725 A190638 A083300 * A223009 A155455 A218349

Adjacent sequences: A308244 A308245 A308246 * A308248 A308249 A308250

Keyword

nonn,more

Author

Esteban Crespi de Valldaura, May 16 2019

Status

approved