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.

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A322666 a(n) is the smallest positive integer k such that there does not exist an m such that floor(m^2/10^n) = k.

Original entry on oeis.org

5, 35, 282, 2600, 25317, 251000, 2503163, 25010000, 250031623, 2500100000, 25000316228, 250001000000, 2500003162278, 25000010000000, 250000031622777, 2500000100000000, 25000000316227767, 250000001000000000, 2500000003162277661, 25000000010000000000
Offset: 1

Views

Author

Jianing Song, Dec 22 2018

Keywords

Comments

For n >= 2, note that when k < 5*10^(n-1) we have (k + 1)^2 - k^2 = 2*k + 1 < 10^n, so there exists m such that floor(m^2/10^n) = 0, 1, 2, ..., 25*10^(n-2). For 0 <= t < sqrt(10^n), floor((5*10^(n-1) + t)^2/10^n) = 25*10^(n-2) + t; for t = ceiling(sqrt(10^n)), floor((5*10^(n-1) + t)^2/10^n) = 25*10^(n-2) + t + 1. So the number 25*10^(n-2) + ceiling(10^(n/2)) is skipped over. Take n = 3 as an example. When k < 500, (k + 1)^2 - k^2 < 1000, so there exists m such that floor(m^2/1000) = 0, 1, 2, ..., 250. Since 31^2 = 961 < 1000, 32^2 = 1024 > 1000, (500 + t)^2 is successively 251001, 252004, ..., 281961, 283024, so a(3) = 282.
The sum of digits for Sum_{i=1..n} a(2*n) is 8*n.

Examples

			floor(m^2/10) = 0, 0, 0, 0, 1, 2, 3, 4, 6 for m = 0..8, so a(1) = 5.
floor(m^2/100) = 0, ..., 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36 for m = 0..60, so a(2) = 35.

Crossrefs

Cf. A017936, A317774, A322667.

Programs

  • PARI
    a(n) = if(n==1, 5, 25*10^(n-2) + ceil(10^(n/2)))

Formula

a(n) = 25*10^(n-2) + ceiling(10^(n/2)) for n >= 2.

A322667 a(n) is the smallest positive integer k such that floor((k + 1)^2/10^n) - floor(k^2/10^n) = 2.

Original entry on oeis.org

7, 59, 531, 5099, 50316, 500999, 5003162, 50009999, 500031622, 5000099999, 50000316227, 500000999999, 5000003162277, 50000009999999, 500000031622776, 5000000099999999, 50000000316227766, 500000000999999999, 5000000003162277660, 50000000009999999999
Offset: 1

Views

Author

Jianing Song, Dec 22 2018

Keywords

Comments

For n >= 2, note that when k < 5*10^(n-1) we have (k + 1)^2 - k^2 = 2*k + 1 < 10^n, so a(n) >= 5*10^(n-1). For 0 <= t < sqrt(10^n), floor((5*10^(n-1) + t)^2/10^n) = 25*10^(n-2) + t; for t = ceiling(sqrt(10^n)), floor((5*10^(n-1) + t)^2/10^n) = 25*10^(n-2) + t + 1. Take n = 3 as an example. When k < 500, (k + 1)^2 - k^2 < 1000, so a(3) >= 500. Since 31^2 = 961 < 1000, 32^2 = 1024 > 1000, (500 + t)^2 is successively 251001, 252004, ..., 281961, 283024, so a(3) = 500 + 31 = 531.

Examples

			floor(7^2/10) = 4, floor(8^2/10) = 6, and 7 is the smallest k such that floor((k + 1)^2/10) - floor(k^2/10) = 2, so a(1) = 7.
floor(59^2/10) = 34, floor(60^2/10) = 36, and 59 is the smallest k such that floor((k + 1)^2/100) - floor(k^2/100) = 2, so a(2) = 59.

Crossrefs

Cf. A017936, A317774, A322666.

Programs

  • PARI
    a(n) = if(n==1, 7, 5*10^(n-1) + ceil(10^(n/2)) - 1)

Formula

a(n) = 5*10^(n-1) + ceiling(10^(n/2)) - 1 for n >= 2.

A130082 Smallest number whose eighth power has at least n digits.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 8, 10, 14, 18, 24, 32, 43, 57, 75, 100, 134, 178, 238, 317, 422, 563, 750, 1000, 1334, 1779, 2372, 3163, 4217, 5624, 7499, 10000, 13336, 17783, 23714, 31623, 42170, 56235, 74990, 100000, 133353, 177828, 237138, 316228, 421697
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2007

Keywords

Comments

Powers of eighth root of 10 rounded up.

Examples

			9^8 = 43046721 has eight digits, 10^8 = 100000000 has nine digits, hence a(9) = 10.

Links

Crossrefs

Cf. A011277, A011557 (powers of 10), A017936 (smallest number whose square has n digits), A018005 (smallest number whose cube has n digits), A018074 (smallest number whose fourth power has n digits), A018143 (smallest number whose fifth power has n digits), A130080 to A130084 (smallest number whose sixth ... tenth power has n digits).

Programs

  • Magma
    [ Ceiling(Root(10^(n-1),8)): n in [1..46] ];
  • Mathematica
    Table[(Ceiling[10^((n - 1)/8)]), {n, 1, 60}] (* Vincenzo Librandi, Sep 20 2013 *)
  • Python
    from sympy import integer_nthroot
def A130082(n): return (lambda x:x[0]+(not x[1]))(integer_nthroot(10**(n-1),8)) # Chai Wah Wu, Jun 20 2024

Formula

a(n) = ceiling(10^((n-1)/8)).

A069658 a(1) = 1; a(n) = smallest nontrivial n-digit perfect power.

Original entry on oeis.org

1, 16, 121, 1024, 10201, 100489, 1002001, 10004569, 100020001, 1000014129, 10000200001, 100000147984, 1000002000001, 10000002149284, 100000020000001, 1000000025191729, 10000000200000001, 100000000621806289, 1000000002000000001
Offset: 1

Views

Author

Amarnath Murthy, Apr 04 2002

Keywords

Comments

Powers of 10 are not allowed.

Crossrefs

Cf. A061432, A017936.

Programs

  • Mathematica
    Join[{1},(Floor[Sqrt[10^Range[20]]]+1)^2] (* Harvey P. Dale, Mar 08 2022 *)

Formula

a(n) = (floor(sqrt(10)^(n-1))+1)^2 = (A017934(n-1)+1)^2. - Vladeta Jovovic, Jun 30 2002

Extensions

More terms from Sascha Kurz, Jan 28 2003

A368088 Index of smallest pentagonal number with n digits.

Original entry on oeis.org

1, 3, 9, 26, 82, 259, 817, 2583, 8166, 25821, 81650, 258200, 816497, 2581990, 8164966, 25819890, 81649659, 258198890, 816496582, 2581988898, 8164965810, 25819888975, 81649658093, 258198889748, 816496580928, 2581988897472, 8164965809278, 25819888974717, 81649658092773
Offset: 1

Views

Author

Kelvin Voskuijl, Dec 17 2023

Keywords

Comments

The digits of the odd- and even-indexed terms converge to those in the decimal expansions of sqrt(2/3) and sqrt(20/3), respectively.

Examples

			a(4) = 26 as the 26th pentagonal number is 26*(3*26-1)/2 = 1001 which has 4 digits (while the 25th is 925 which is only 3).

Crossrefs

Cf. A000326, A180447.
Cf. A068092 (for triangular numbers), A017936 (for squares).
Cf. A157697 (square root of 2/3), A020772 (square root of 20/3)

Programs

  • Mathematica
    a[n_] := Ceiling[(Sqrt[24*10^(n-1) + 1] + 1)/6]; Array[a, 40] (* Amiram Eldar, Dec 30 2023 *)
  • PARI
    a(n) = 1 + (sqrtint(24*10^(n-1)) + 1)\6 \\ Andrew Howroyd, Dec 30 2023

Formula

a(n) = ceiling((sqrt(24*10^(n-1) + 1) + 1)/6).
Previous Showing 11-15 of 15 results.

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