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

%I A326319 #21 Aug 23 2019 13:56:37
%S A326319 9007,10091,10531,10831,11801,12197,12431,12833,12941,13393,13501,
%T A326319 13619,13679,13751,13907,13939,14219,14423,14737,14851,14893,15217,
%U A326319 15641,15767,15773,15803,15959,16019,16201,16241,16393,16397,16417,16441,16517,16559
%N A326319 Numbers that cannot be written as a difference of 11-smooth numbers.
%C A326319 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:
%C A326319   <2> +- k !== <3, 5, 7, 11> mod 563213996185633,
%C A326319   <3> +- k !== <2, 5, 7, 11> mod 194191394486113583,
%C A326319   <2, 3> +- k !== <5, 7, 11> mod 1762314762258271,
%C A326319   <5> +- k !== <2, 3, 7, 11> mod 220836983154619,
%C A326319   <2, 5> +- k !== <3, 7, 11> mod 2128827364461031,
%C A326319   <3, 5> +- k !== <2, 7, 11> mod 3521575252831519,
%C A326319   <7, 11> +- k !== <2, 3, 5> mod 497846284658749,
%C A326319   <7> +- k !== <2, 3, 5, 11> mod 5489574535421899,
%C A326319   <2, 7> +- k !== <3, 5, 11> mod 6600281111334703,
%C A326319   <3, 7> +- k !== <2, 5, 11> mod 834486158701066937,
%C A326319   <5, 11> +- k !== <2, 3, 7> mod 239190476358328703,
%C A326319   <5, 7> +- k !== <2, 3, 11> mod 3288443009987083,
%C A326319   <3, 11> +- k !== <2, 5, 7> mod 14071029652900961,
%C A326319   <2, 11> +- k !== <3, 5, 7> mod 1762314762258271,
%C A326319   <11> +- k !== <2, 3, 5, 7> mod 411934385702047,
%C A326319 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.
%H A326319 Esteban Crespi de Valldaura, <a href="/A326319/b326319.txt">Table of n, a(n) for n = 1..101</a>
%e A326319 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.
%Y A326319 Cf. A051038 (11-smooth numbers).
%Y A326319 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).
%Y A326319 Cf. A308247.
%K A326319 nonn
%O A326319 1,1
%A A326319 _Esteban Crespi de Valldaura_, Jun 26 2019