A343104 Smallest number having exactly n divisors of the form 8*k + 1.

1, 9, 81, 153, 891, 1377, 8019, 3825, 11025, 15147, 88209, 31977, 354375, 99225, 121275, 95931, 7144929, 187425, 893025, 287793, 1403325, 1499553, 1715175, 675675, 1091475, 6024375, 1576575, 1686825, 72335025, 2027025, 2264802453041139, 2297295, 11609325, 121463793, 9823275

Offset

1,2

Comments

Smallest index of n in A188169.

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).

From David A. Corneth, Apr 05 2021: (Start)

All terms are odd since if a term is even then the odd part has the same number of such divisors.

No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).

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).

If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)

Links

Table of n, a(n) for n=1..35.

Formula

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.

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.

Example

a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.

Prog

(PARI) res(n, a, b) = sumdiv(n, d, (d%a) == b)
a(n) = if(n>0, for(k=1, oo, if(res(k, 8, 1)==n, return(k))))

Crossrefs

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).

Cf. A188169.

Cf. A007519, A007520, A007521, A007522.

Sequence in context: A207591 A207924 A343134 * A043180 A207875 A207678

Adjacent sequences: A343101 A343102 A343103 * A343105 A343106 A343107

Keyword

nonn

Author

Jianing Song, Apr 05 2021

Extensions

More terms from David A. Corneth, Apr 06 2021

Status

approved