A326319 Numbers that cannot be written as a difference of 11-smooth numbers.

9007, 10091, 10531, 10831, 11801, 12197, 12431, 12833, 12941, 13393, 13501, 13619, 13679, 13751, 13907, 13939, 14219, 14423, 14737, 14851, 14893, 15217, 15641, 15767, 15773, 15803, 15959, 16019, 16201, 16241, 16393, 16397, 16417, 16441, 16517, 16559

Offset: 1

Author

Esteban Crespi de Valldaura, Jun 26 2019

Keywords

• nonn

Comments

Terms were found by generating in sequential order the 11-smooth numbers up to some limit and collecting the differences. The first 100 candidates k were then proved to be correct by showing that each of the following 15 congruences holds:

<2> +- k !== <3, 5, 7, 11> mod 563213996185633,

<3> +- k !== <2, 5, 7, 11> mod 194191394486113583,

<2, 3> +- k !== <5, 7, 11> mod 1762314762258271,

<5> +- k !== <2, 3, 7, 11> mod 220836983154619,

<2, 5> +- k !== <3, 7, 11> mod 2128827364461031,

<3, 5> +- k !== <2, 7, 11> mod 3521575252831519,

<7, 11> +- k !== <2, 3, 5> mod 497846284658749,

<7> +- k !== <2, 3, 5, 11> mod 5489574535421899,

<2, 7> +- k !== <3, 5, 11> mod 6600281111334703,

<3, 7> +- k !== <2, 5, 11> mod 834486158701066937,

<5, 11> +- k !== <2, 3, 7> mod 239190476358328703,

<5, 7> +- k !== <2, 3, 11> mod 3288443009987083,

<3, 11> +- k !== <2, 5, 7> mod 14071029652900961,

<2, 11> +- k !== <3, 5, 7> mod 1762314762258271,

<11> +- k !== <2, 3, 5, 7> mod 411934385702047,

where represents any element in the subgroup generated by a,b,... of the multiplicative subgroup modulo m. For a discussion iof this method of proof see A308247.

Examples

			9007 = A308247(5) cannot be written as the difference of 11-smooth numbers. All smaller numbers can; for example, 1849 = 3^4*5^2 - 2^4*11, 2309 = 2*3^5*5 - 11^2.
		

Links

• Esteban Crespi de Valldaura, Table of n, a(n) for n = 1..101

Crossrefs

Cf. A051038 (11-smooth numbers).

Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A308456 (p=5), A326318 (p=7), A326320 (p=13).

Cf. A308247.