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

%I A326318 #15 Aug 23 2019 13:35:20
%S A326318 1849,2309,2411,2483,2507,2531,2629,2711,2753,2843,2851,2921,2941,
%T A326318 3139,3161,3167,3181,3217,3229,3251,3287,3289,3293,3323,3379,3481,
%U A326318 3487,3541,3601,3623,3653,3697,3698,3709,3737,3739,3803,3827,3859,3877,3901,3923,3947
%N A326318 Numbers that cannot be written as a difference of 7-smooth numbers (A002473).
%C A326318 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:
%C A326318    <2> +- k !== <3, 5, 7> mod 31487336959,
%C A326318    <3> +- k !== <2, 5, 7> mod 121328339431,
%C A326318    <2, 3> +- k !== <5, 7> mod 5699207989579,
%C A326318    <5> +- k !== <2, 3, 7> mod 1206047658673,
%C A326318    <2, 5> +- k !== <3, 7> mod 11174958041,
%C A326318    <3, 5> +- k !== <2, 7> mod 31487336959,
%C A326318    <7> +- k !== <2, 3, 5> mod 1116870318707,
%C A326318 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 A326318 Esteban Crespi de Valldaura, <a href="/A326318/b326318.txt">Table of n, a(n) for n = 1..101</a>
%e A326318 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.
%Y A326318 Cf. A002473 (7-smooth numbers).
%Y A326318 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).
%Y A326318 Cf. A308247.
%K A326318 nonn
%O A326318 1,1
%A A326318 _Esteban Crespi de Valldaura_, Jun 26 2019