A326318 Numbers that cannot be written as a difference of 7-smooth numbers (A002473).

1849, 2309, 2411, 2483, 2507, 2531, 2629, 2711, 2753, 2843, 2851, 2921, 2941, 3139, 3161, 3167, 3181, 3217, 3229, 3251, 3287, 3289, 3293, 3323, 3379, 3481, 3487, 3541, 3601, 3623, 3653, 3697, 3698, 3709, 3737, 3739, 3803, 3827, 3859, 3877, 3901, 3923, 3947

Offset: 1

Author

Esteban Crespi de Valldaura, Jun 26 2019

Keywords

• nonn

Comments

Terms were found by generating in sequential order the 7-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 congruences holds:

<2> +- k !== <3, 5, 7> mod 31487336959,

<3> +- k !== <2, 5, 7> mod 121328339431,

<2, 3> +- k !== <5, 7> mod 5699207989579,

<5> +- k !== <2, 3, 7> mod 1206047658673,

<2, 5> +- k !== <3, 7> mod 11174958041,

<3, 5> +- k !== <2, 7> mod 31487336959,

<7> +- k !== <2, 3, 5> mod 1116870318707,

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

			1849 = A308247(4) cannot be written as the difference of 7-smooth numbers. All smaller numbers can; for example, 281 = 2^5*3^2 - 7, 289 = 2*3*7^2 - 5, ..., 1847 = 3*5^4 - 2^2*7.
		

Links

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

Crossrefs

Cf. A002473 (7-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), A326319 (p=11), A326320 (p=13).

Cf. A308247.