

A343106


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


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
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OFFSET

0,2


COMMENTS

Smallest index of n in A188171.
a(n) exists for all n, since 5*3^(2n2) has exactly n divisors of the form 8*k + 1, namely 5*3^0, 5*3^2, ..., 5*3^(2n2). This actually gives an upper bound (which is too far from reality when n is large) for a(n).


LINKS

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


FORMULA

a(2n1) <= 3^(2n2) * 35, since 3^(2n2) * 35 has exactly 2n1 divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^(2n2) * 5, 3^1 * 7, 3^3 * 7, ..., 3^(2n3) * 7.
a(2n) <= 3^(n1) * 455, since 3^(n1) * 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 <= n1 and b is the largest even number <= n1.


EXAMPLE

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


PROG

(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)))


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.
Sequence in context: A303408 A241275 A343136 * A081070 A247494 A043025
Adjacent sequences: A343103 A343104 A343105 * A343107 A343108 A343109


KEYWORD

nonn


AUTHOR

Jianing Song, Apr 05 2021


EXTENSIONS

More terms from Bert Dobbelaere, Apr 09 2021


STATUS

approved



