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

1,1

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 <a,b,...> 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.

Links

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

Example

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.

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.

Sequence in context: A247724 A236030 A237892 * A115719 A062670 A206468

Adjacent sequences: A326315 A326316 A326317 * A326319 A326320 A326321

Keyword

nonn

Author

Esteban Crespi de Valldaura, Jun 26 2019

Status

approved