A067497 -id:A067497 - cp's OEIS frontend

A367295 Numbers k such that 6 is the first digit of 2^k.

6, 16, 26, 36, 79, 89, 99, 109, 119, 129, 139, 182, 192, 202, 212, 222, 232, 275, 285, 295, 305, 315, 325, 335, 368, 378, 388, 398, 408, 418, 428, 471, 481, 491, 501, 511, 521, 564, 574, 584, 594, 604, 614, 624, 667, 677, 687, 697, 707, 717, 760, 770, 780, 790

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(7/6) = 0.066946...

Cf. A000079, A067497, A067469, A172404, A367294, A363060, A330243, A367296, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 6 <= x and x < 7 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;

Select[Range[800], IntegerDigits[2^#][[1]] == 6 &] (* Amiram Eldar, Nov 12 2023 *)

A198266 a(n) = ceiling(n*sqrt(11)).

Original entry on oeis.org

0, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 73, 77, 80, 83, 87, 90, 93, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 136, 140, 143, 146, 150, 153, 156, 160, 163, 166, 170, 173, 176, 180

Offset: 0

Vincenzo Librandi, Oct 24 2011

The first term where this sequence is different from A067497 is a(22)=73, compared with A067497(22)=74. - Jean-François Alcover, Jan 29 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A067497, A171982.

Magma
```
[Ceiling(n*Sqrt(11)): n in [0..60]]
```

Ceiling[Range[0,60]Sqrt[11]] (* Harvey P. Dale, Aug 20 2017 *)

A198267 Round(n*sqrt(11)).

Original entry on oeis.org

0, 3, 7, 10, 13, 17, 20, 23, 27, 30, 33, 36, 40, 43, 46, 50, 53, 56, 60, 63, 66, 70, 73, 76, 80, 83, 86, 90, 93, 96, 99, 103, 106, 109, 113, 116, 119, 123, 126, 129, 133, 136, 139, 143, 146, 149, 153, 156, 159, 163, 166, 169, 172, 176, 179

Offset: 0

Vincenzo Librandi, Oct 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A067497, A171982.

Magma
```
[Round(n*Sqrt(11)): n in [0..60]]
```

A361977 a(n) is the largest prime p such that 2^p - 1 <= 10^n.

Original entry on oeis.org

3, 5, 7, 13, 13, 19, 23, 23, 29, 31, 31, 37, 43, 43, 47, 53, 53, 59, 61, 61, 67, 73, 73, 79, 83, 83, 89, 89, 89, 97, 101, 103, 109, 109, 113, 113, 113, 113, 127, 131, 131, 139, 139, 139, 149, 151, 151, 157, 157, 163, 167, 167, 173, 179, 181, 181, 181, 191, 193

Offset: 1

Jack Braxton, Apr 02 2023

			a(5) = 13, 2^13 - 1 = 8192 < 10^5.

Cf. A000040, A000043, A000668, A011557, A067497, A151799.

from sympy import prevprime
def A361977(n): return prevprime((10**n-1).bit_length()) # Chai Wah Wu, Apr 02 2023

a(n) = A151799(A067497(n)). - Alois P. Heinz, Apr 02 2023

cp's OEIS Frontend

A367295 Numbers k such that 6 is the first digit of 2^k.

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A198266 a(n) = ceiling(n*sqrt(11)).

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Magma

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A198267 Round(n*sqrt(11)).

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Magma

A361977 a(n) is the largest prime p such that 2^p - 1 <= 10^n.

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Python

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