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A343105 Smallest number having exactly n divisors of the form 8*k + 3.

%I A343105 #12 Apr 09 2021 03:50:02
%S A343105 1,3,27,99,297,891,1683,8019,5049,17325,15147,99225,31977,190575,
%T A343105 136323,121275,95931,3189375,225225,64304361,287793,1289925,1686825,
%U A343105 15526875,675675,1091475,3239775,1576575,2590137,251644717004571,2027025,15436575,2297295,28676025,33350625,9823275,3828825,42879375,760816875
%N A343105 Smallest number having exactly n divisors of the form 8*k + 3.
%C A343105 Smallest index of n in A188170.
%C A343105 a(n) exists for all n, since 3^(2n-1) has exactly n divisors of the form 8*k + 3, namely 3^1, 3^3, ..., 3^(2n-1). This actually gives an upper bound (which is too far from reality when n is large) for a(n).
%H A343105 Bert Dobbelaere, <a href="/A343105/b343105.txt">Table of n, a(n) for n = 0..200</a>
%F A343105 a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 3 modulo 8: 3^1, 3^3, ..., 3^(2n-3), 3^0 * 11, 3^2 * 11, ..., 3^(2n-2) * 11.
%F A343105 a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 3 modulo 8: 3^1, 3^3, ..., 3^a, 3^1 * 17, 3^3 * 17, ..., 3^a * 17, 3^0 * 11, 3^2 * 11, ..., 3^b * 11, 3^0 * 187, 3^2 * 187, ... 3^b * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.
%e A343105 a(4) = 297 since it is the smallest number with exactly 4 divisors congruent to 3 modulo 8, namely 3, 11, 27 and 297.
%o A343105 (PARI) res(n,a,b) = sumdiv(n, d, (d%a) == b)
%o A343105 a(n) = for(k=1, oo, if(res(k,8,3)==n, return(k)))
%Y A343105 Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), this sequence (i=3), A343106 (i=5), A188226 (i=7).
%Y A343105 Cf. A188170.
%K A343105 nonn
%O A343105 0,2
%A A343105 _Jianing Song_, Apr 05 2021
%E A343105 More terms from _Bert Dobbelaere_, Apr 09 2021