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A343106 Smallest number having exactly n divisors of the form 8*k + 5.

%I A343106 #11 Apr 09 2021 03:49:35
%S A343106 1,5,45,315,585,2205,2925,14175,9945,17325,28665,178605,45045,190575,
%T A343106 240975,143325,135135,3189375,225225,93002175,405405,1403325,1715175,
%U A343106 2401245,675675,3583125,3239775,1576575,3468465,94918019805,2027025,15436575,2297295,11609325,16769025,27286875,3828825,42879375,117661005
%N A343106 Smallest number having exactly n divisors of the form 8*k + 5.
%C A343106 Smallest index of n in A188171.
%C A343106 a(n) exists for all n, since 5*3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 5*3^0, 5*3^2, ..., 5*3^(2n-2). This actually gives an upper bound (which is too far from reality when n is large) for a(n).
%H A343106 Bert Dobbelaere, <a href="/A343106/b343106.txt">Table of n, a(n) for n = 0..200</a>
%F A343106 a(2n-1) <= 3^(2n-2) * 35, since 3^(2n-2) * 35 has exactly 2n-1 divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^(2n-2) * 5, 3^1 * 7, 3^3 * 7, ..., 3^(2n-3) * 7.
%F A343106 a(2n) <= 3^(n-1) * 455, since 3^(n-1) * 455 has exactly 2n divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^b * 5, 3^1 * 7, 3^3 * 7, ..., 3^a * 7, 3^0 * 13, 3^2 * 13, ..., 3^b * 13, 3^1 * 455, 3^3 * 455, ... 3^a * 455, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.
%e A343106 a(4) = 585 since it is the smallest number with exactly 4 divisors congruent to 5 modulo 8, namely 5, 13, 45 and 585.
%o A343106 (PARI) res(n,a,b) = sumdiv(n, d, (d%a) == b)
%o A343106 a(n) = for(k=1, oo, if(res(k,8,5)==n, return(k)))
%Y A343106 Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), this sequence (i=5), A188226 (i=7).
%Y A343106 Cf. A188171.
%K A343106 nonn
%O A343106 0,2
%A A343106 _Jianing Song_, Apr 05 2021
%E A343106 More terms from _Bert Dobbelaere_, Apr 09 2021