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A326319
Numbers that cannot be written as a difference of 11-smooth numbers.
5
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
1,1
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 <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
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.
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.
Sequence in context: A215198 A202531 A106770 * A102484 A295457 A359845
KEYWORD
nonn
AUTHOR
STATUS
approved