

A308247


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


5




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 3smooth is enough to prove that 41 is not the difference of 3smooth 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 5smooth is enough to show that 281 is not the difference of 5smooth 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 ABCtriples 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 = 21, 2 = 42, 3 = 41, and 4 = 84. 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 3smooth numbers, but as shown above 41 is not, so a(2)=41.


CROSSREFS

Psmooth_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



