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

1, 5, 45, 315, 585, 2205, 2925, 14175, 9945, 17325, 28665, 178605, 45045, 190575, 240975, 143325, 135135, 3189375, 225225, 93002175, 405405, 1403325, 1715175, 2401245, 675675, 3583125, 3239775, 1576575, 3468465, 94918019805, 2027025, 15436575, 2297295, 11609325, 16769025, 27286875, 3828825, 42879375, 117661005

Offset: 0

Author

Jianing Song, Apr 05 2021

Keywords

• nonn

Comments

Smallest index of n in A188171.

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).

Examples

			a(4) = 585 since it is the smallest number with exactly 4 divisors congruent to 5 modulo 8, namely 5, 13, 45 and 585.
		

Links

• Bert Dobbelaere, Table of n, a(n) for n = 0..200

Crossrefs

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).

Cf. A188171.

Programs

PARI

res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = for(k=1, oo, if(res(k,8,5)==n, return(k)))

Formula

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.

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.

Extensions

More terms from Bert Dobbelaere, Apr 09 2021