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%I #22 Apr 09 2021 16:40:37
%S 1,9,81,153,891,1377,8019,3825,11025,15147,88209,31977,354375,99225,
%T 121275,95931,7144929,187425,893025,287793,1403325,1499553,1715175,
%U 675675,1091475,6024375,1576575,1686825,72335025,2027025,2264802453041139,2297295,11609325,121463793,9823275
%N Smallest number having exactly n divisors of the form 8*k + 1.
%C Smallest index of n in A188169.
%C a(n) exists for all n, since 3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 3^0, 3^2, ..., 3^(2n-2). This actually gives an upper bound for a(n).
%C From _David A. Corneth_, Apr 05 2021: (Start)
%C All terms are odd since if a term is even then the odd part has the same number of such divisors.
%C No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).
%C If for some k, A188169(k) > m then A188169(k*t) > m for all t > 0. This can be used to trim searches when looking for some a(m).
%C If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)
%F a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^(2n-2), 3^1 * 11, 3^3 * 11, ..., 3^(2n-3) * 11.
%F a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^b, 3^0 * 17, 3^2 * 17, ..., 3^b * 17, 3^1 * 11, 3^3 * 11, ..., 3^a * 11, 3^1 * 187, 3^3 * 187, ... 3^a * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.
%e a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.
%o (PARI) res(n,a,b) = sumdiv(n, d, (d%a) == b)
%o a(n) = if(n>0, for(k=1, oo, if(res(k,8,1)==n, return(k))))
%Y Smallest number having exactly n divisors of the form 8*k + i: this sequence (i=1), A343105 (i=3), A343106 (i=5), A188226 (i=7).
%Y Cf. A188169.
%Y Cf. A007519, A007520, A007521, A007522.
%K nonn
%O 1,2
%A _Jianing Song_, Apr 05 2021
%E More terms from _David A. Corneth_, Apr 06 2021
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