A102572 -id:A102572 - cp's OEIS frontend

A212448 Floor(4n + log(4n)).

5, 10, 14, 18, 22, 27, 31, 35, 39, 43, 47, 51, 55, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212449, A212450, A212451, A212452, A212453, A212454.

[Floor(4*n + Log(4*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Floor[4*n + Log[4*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

a(n)=4*n+log(4*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212449 Floor(5n + log(5n)).

Original entry on oeis.org

6, 12, 17, 22, 28, 33, 38, 43, 48, 53, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212450, A212451, A212452, A212453, A212454.

[Floor(5*n + Log(5*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Floor[5*n + Log[5*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

a(n)=5*n+log(5*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212450 a(n) = ceiling(n + log(n)).

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212449, A212451, A212452, A212453, A212454.

[Ceiling(n + Log(n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[n + Log[n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

A374056 a(n) = max_{i=0..n} S_4(i) + S_4(n-i) where S_4(x) = A053737(x) is the base-4 digit sum of x.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14

Offset: 0

Hermann Gruber, Jun 26 2024

As shown in the proof of [Gruber and Holzer, lemma 9], the maximum is attained by choosing i as the largest number not exceeding n whose ternary representation is (33...3)_4. Also by lemma 6, for this choice of i we have A053737(i) = 3*floor(log_4(n+1)) and A053737(n-i) = A053737(n+1)-1, giving the formula below.

			For n=74, the maximum is attained by 63 + 11 = (333)_4 + (23)_4. Using 75=(1023)_4, comparing with the formula above, A053737(63) = 3*floor(log_4(n+1)) = 9 and A053737(11) = A053737(74+1)-1 = 5. Notice that other pairs attain the maximum as well. Namely, 43 + 31 = (223)_4 + (133)_4, as well as 47 + 27 = (233)_4 + (123)_4, and 59 + 15 = (323)_4 + (33)_4.

Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length. Extended journal version, in preparation, 2024.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length, Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science, Article No. 53, pp. 53:1-53:15, 2021.

Cf. A053737, A102572, A014701, A374054.

Table[3*Floor[Log[4, k]] + DigitSum[k, 4] - 1, {k, 100}] (* Paolo Xausa, Aug 01 2024 *)

a(n) = 3*logint(n+1, 4) + sumdigits(n+1, 4) - 1;

a(n) = 3*floor(log_4(n+1)) + A053737(n+1) - 1 [Gruber and Holzer, lemma 9].

A363827 Highest power of 2 dividing n which is <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4

Offset: 1

Ilya Gutkovskiy, Oct 19 2023

Cf. A006519, A033676, A135517, A363828.

Cf. A007814, A102572.

Table[Last[Select[Divisors[n], # <= Sqrt[n] && IntegerQ[Log[2, #]] &]], {n, 100}]
a[n_] := 2^Min[IntegerExponent[n, 2], Floor[Log2[n]/2]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

a(n) = if (n==1, 1, vecmax(select(x->((x^2 <= n) && (2^logint(x,2)==x)), divisors(n)))); \\ Michel Marcus, Oct 19 2023

a(n) = 2^min(A007814(n), A102572(n)). - Kevin Ryde, Oct 20 2023

cp's OEIS Frontend

A212448 Floor(4n + log(4n)).

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A212449 Floor(5n + log(5n)).

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A212450 a(n) = ceiling(n + log(n)).

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A374056 a(n) = max_{i=0..n} S_4(i) + S_4(n-i) where S_4(x) = A053737(x) is the base-4 digit sum of x.

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A363827 Highest power of 2 dividing n which is <= sqrt(n).

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