cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A343105 Smallest number having exactly n divisors of the form 8*k + 3.

Original entry on oeis.org

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
Offset: 0

Views

Author

Jianing Song, Apr 05 2021

Keywords

Comments

Smallest index of n in A188170.
a(n) exists for all n, since 3^(2n-1) has exactly n divisors of the form 8*k + 3, namely 3^1, 3^3, ..., 3^(2n-1). This actually gives an upper bound (which is too far from reality when n is large) for a(n).

Examples

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

Links

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.

Programs

  • 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)))

Formula

a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 3 modulo 8: 3^1, 3^3, ..., 3^(2n-3), 3^0 * 11, 3^2 * 11, ..., 3^(2n-2) * 11.
a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 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 <= n-1 and b is the largest even number <= n-1.

Extensions

More terms from Bert Dobbelaere, Apr 09 2021

A188226 Smallest number having exactly n divisors of the form 8*k + 7.

Original entry on oeis.org

1, 7, 63, 315, 945, 1575, 3465, 19845, 10395, 17325, 26775, 127575, 45045, 266805, 190575, 155925, 135135, 2480625, 225225, 130203045, 405405, 1289925, 2168775, 1715175, 675675, 3898125, 3468465, 1576575, 3239775, 67798585575, 2027025, 16769025, 2297295, 20539575, 42170625, 27286875, 3828825, 117661005
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 26 2011

Keywords

Comments

A188172(a(n)) = n and A188172(m) <> n for m < a(n).

Links

Crossrefs

Cf. A004771, A141164, A188172.
Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), A343106 (i=5), this sequence (i=7).

Programs

  • Haskell
    import Data.List  (elemIndex)
import Data.Maybe (fromJust)
a188226 n = a188226_list !! n
a188226_list =
   map (succ . fromJust . (`elemIndex` (map a188172 [1..]))) [0..]

Extensions

a(19)-a(35) from Nathaniel Johnston, Apr 06 2011
More terms from Bert Dobbelaere, Apr 09 2021

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Apr 05 2021

Keywords

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)

Examples

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

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.

Programs

  • 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))))

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.

Extensions

More terms from David A. Corneth, Apr 06 2021

A343136 Positions of records in A188171.

Original entry on oeis.org

1, 5, 45, 315, 585, 2205, 2925, 9945, 17325, 28665, 45045, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815, 218243025, 509233725, 654729075, 1003917915, 1527701175, 3011753745, 4583103525
Offset: 1

Views

Author

David A. Corneth, Apr 06 2021

Keywords

Examples

			585 is in the sequence as A188171(585) = 4 via the divisors 5, 13, 45 and 117 and no positive integer < 585 has at least four such (5 (mod 8)) divisors.

Crossrefs

Cf. A188171, A343106.

Programs

  • Mathematica
    Function[{s, r}, Map[FirstPosition[s, #][[1]] &, r]] @@ {#, Union@ FoldList[Max, #]} &@ Array[DivisorSum[#, 1 &, Mod[#, 8] == 5 &] &, 10^6] (* Michael De Vlieger, Apr 08 2021 *)
Showing 1-4 of 4 results.

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