A281149 -id:A281149 - cp's OEIS frontend

A281150 Elias delta code for n.

1, 1000, 1001, 10100, 10101, 10110, 10111, 11000000, 11000001, 11000010, 11000011, 11000100, 11000101, 11000110, 11000111, 110010000, 110010001, 110010010, 110010011, 110010100, 110010101, 110010110, 110010111, 110011000, 110011001, 110011010

Offset: 1

Indranil Ghosh, Jan 16 2017

The number of bits in a(n) is equal to A140341(n).

a(n) is the prefix-free encoding of n-1 defined on pages 180-181 of Shallit (2008). - N. J. A. Sloane, Mar 18 2019

			For n = 9, the first part is "11000" and the second part is "001". So a(9) = 11000001.

Shallit, Jeffrey. A second course in formal languages and automata theory. Cambridge University Press, 2008. See E(m) on page 181. - N. J. A. Sloane, Mar 18 2019

Indranil Ghosh, Table of n, a(n) for n = 1..10000
J. Nelson Raja, P. Jaganathan and S. Domnic, A New Variable-Length Integer Code for Integer Representation and Its Application to Text Compression, Indian Journal of Science and Technology, Vol 8(24), September 2015.

Cf. A140341, A281149.

Unary(n) = A105279(n-1).

import math
def unary(n):
    return "1"*(n-1)+"0"
def elias_gamma(n):
    if n==1:
        return "1"
    k=int(math.log(n, 2))
    fp=unary(1+k)    #fp is the first part
    sp=n-2**(k)      #sp is the second part
    nb=k             #nb is the number of bits used to store sp in binary
    sp=bin(sp)[2:]
    if len(sp)

For a given integer n, a(n) is composed of two parts. The first part equals 1+floor(log_2 n) and the second part equals n-2^(floor(log_2 n)). The first part is stored in Elias Gamma Code and the second part is stored in a binary using floor(log_2 n) bits. The first and the second parts are concatenated to give a(n).

A129972 a(n) = 2*floor(log_2(n)) + 1.

Original entry on oeis.org

1, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Benoit Cloitre, Jun 13 2007

The number of bits needed to write n using Elias gamma coding. - Charles R Greathouse IV, Mar 21 2012

Consists of the n-th odd number (A005408(n) = 2n+1) repeated 2^(n-1) times (since a(n) = a(n-1) except when a(n) > a(n-1) which happens for n a power of 2). - Jonathan Vos Post, Jun 17 2007

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A005408.

Cf. A281149 (See the comment section by Charles R Greathouse IV, of this sequence (A129972) ). - Indranil Ghosh, Jan 17 2017

PARI
```
a(n)=2*floor(log(n)/log(2))+1
```

a(n)=log(n+.5)\log(2)*2+1 \\ Charles R Greathouse IV, Mar 21 2012

a(n)=2*logint(n,2)+1 \\ Charles R Greathouse IV, Sep 04 2015

a(n) = Sum_{k=1..n} (-1)^(k-1)*floor(n/k)*mu(k).

A281193 Elias's omega code for n.

Original entry on oeis.org

0, 100, 110, 101000, 101010, 101100, 101110, 1110000, 1110010, 1110100, 1110110, 1111000, 1111010, 1111100, 1111110, 10100100000, 10100100010, 10100100100, 10100100110, 10100101000, 10100101010, 10100101100, 10100101110, 10100110000, 10100110010, 10100110100

Offset: 1

Indranil Ghosh, Jan 17 2017

The idea of the Elias omega code is similar to that of the Elias delta code (A281150), except that the length of the codeword in the omega code is recursively encoded.

The number of bits in a(n) is equal to A072464(n).

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Khalid Sayood (Editor), Lossless Compression Handbook, Chapter 3 - Universal Codes, p. 59, section 3.6.
Wikipedia, Elias omega coding

Cf. A072464, A281149, A281150.

def E(n):
    s=""
    if n==1:
        return "0"
    else:
        b=(bin(n)[2:])
        s+=E(len(b)-1)+b
    return s
def elias_omega(n):
    return int(E(n)[1:]+"0")

A281551 Prime numbers p such that the decimal representation of its Elias gamma code is also a prime.

Original entry on oeis.org

3, 23, 41, 47, 59, 89, 101, 149, 179, 227, 317, 347, 353, 383, 389, 479, 503, 599, 821, 887, 929, 977, 1019, 1109, 1229, 1283, 1319, 1511, 1571, 1619, 1667, 1709, 1733, 1787, 1847, 1889, 1907, 1913, 1931, 2207, 2309, 2333, 2357, 2399, 2417, 2459, 2609, 2753, 2789, 2909, 2963, 2999, 3203, 3257, 3299

Offset: 1

Indranil Ghosh, Jan 24 2017

			59 is in the sequence because the decimal representation of its Elias gamma code is 2011 and both 59 and 2011 are prime numbers.

Indranil Ghosh, Table of n, a(n) for n = 1..2014

Cf. A000040, A171885 (decimal representation of Elias gamma code), A281149, A281316.

import math
from sympy import isprime
def unary(n):
    return "1"*(n-1)+"0"
def elias_gamma(n):
    if n ==1:
        return "1"
    k=int(math.log(n,2))
    fp=unary(1+k)    #fp is the first part
    sp=n-2**(k)      #sp is the second part
    nb=k             #nb is the number of bits used to store sp in binary
    sp=bin(sp)[2:]
    if len(sp)

A281552 Write n in the Elias gamma code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 6, 2, 3, 3, 9, 3, 8, 8, 9, 3, 4, 4, 12, 4, 11, 11, 12, 4, 10, 10, 18, 10, 11, 11, 12, 4, 5, 5, 15, 5, 14, 14, 15, 5, 13, 13, 23, 13, 14, 14, 15, 5, 12, 12, 22, 12, 21, 21, 22, 12, 13, 13, 23, 13, 14, 14, 15, 5, 6, 6, 18, 6, 17, 17, 18, 6, 16, 16, 28, 16, 17, 17, 18, 6

Offset: 1

Indranil Ghosh, Jan 24 2017

			For n = 6 , the Elias gamma code for n is '11010'. In '11010', the positions of '1' followed immediately to the right by '0' counting from left are 2 and 4. So, a(6) = 2 + 4 = 6.
For n = 10, the Elias gamma code for n is '1110010'. In '1110010', the positions of '1' followed immediately to the right by '0' counting from left are 3 and 6. So, a(10) = 3 + 6 = 9.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A049501, A171885, A281149, A281388, A281497.

def a(n):
    x= A281149(n)
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049501(A171885(n)) for n > = 1.

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A281150 Elias delta code for n.

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A129972 a(n) = 2*floor(log_2(n)) + 1.

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A281193 Elias's omega code for n.

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A281551 Prime numbers p such that the decimal representation of its Elias gamma code is also a prime.

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Python

A281552 Write n in the Elias gamma code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

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