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

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Apr 05 2021

Keywords

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

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

A343134 Positions of records in A188169.

Original entry on oeis.org

1, 9, 81, 153, 891, 1377, 3825, 11025, 15147, 31977, 95931, 187425, 287793, 675675, 1091475, 1576575, 1686825, 2027025, 2297295, 3828825, 6185025, 11486475, 18555075, 26801775, 34459425, 43648605, 72747675, 117515475, 218243025, 352546425, 509233725, 654729075, 1003917915
Offset: 1

Views

Author

David A. Corneth, Apr 06 2021

Keywords

Examples

			153 is in the sequence as A188169(153) = 4 via the divisors 1, 9, 17 and 153 and no positive integer < 153 has at least four such (1 (mod 8)) divisors.

Crossrefs

Cf. A188169, A343104.
Showing 1-4 of 4 results.

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