

A343105


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


5



1, 3, 27, 99, 297, 891, 1683, 8019, 5049, 17325, 15147, 99225, 31977, 190575, 136323, 121275, 95931, 3189375, 225225, 64304361, 287793, 1289925, 1686825, 15526875, 675675, 1091475, 3239775, 1576575, 2590137, 251644717004571, 2027025, 15436575, 2297295, 28676025, 33350625, 9823275, 3828825, 42879375, 760816875
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Smallest index of n in A188170.
a(n) exists for all n, since 3^(2n1) has exactly n divisors of the form 8*k + 3, namely 3^1, 3^3, ..., 3^(2n1). 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) * 11, since 3^(2n2) * 11 has exactly 2n1 divisors congruent to 3 modulo 8: 3^1, 3^3, ..., 3^(2n3), 3^0 * 11, 3^2 * 11, ..., 3^(2n2) * 11.
a(2n) <= 3^(n1) * 187, since 3^(n1) * 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 <= n1 and b is the largest even number <= n1.


EXAMPLE

a(4) = 297 since it is the smallest number with exactly 4 divisors congruent to 3 modulo 8, namely 3, 11, 27 and 297.


PROG

(PARI) res(n, a, b) = sumdiv(n, d, (d%a) == b)
a(n) = for(k=1, oo, if(res(k, 8, 3)==n, return(k)))


CROSSREFS

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).
Cf. A188170.
Sequence in context: A045491 A318908 A343135 * A200977 A302525 A303407
Adjacent sequences: A343102 A343103 A343104 * A343106 A343107 A343108


KEYWORD

nonn


AUTHOR

Jianing Song, Apr 05 2021


EXTENSIONS

More terms from Bert Dobbelaere, Apr 09 2021


STATUS

approved



