A171885 -id:A171885 - cp's OEIS frontend

A281149 Elias gamma code (EGC) for n.

1, 100, 101, 11000, 11001, 11010, 11011, 1110000, 1110001, 1110010, 1110011, 1110100, 1110101, 1110110, 1110111, 111100000, 111100001, 111100010, 111100011, 111100100, 111100101, 111100110, 111100111, 111101000, 111101001, 111101010, 111101011, 111101100, 111101101, 111101110

Offset: 1

Indranil Ghosh, Jan 16 2017

This sequence is the binary equivalent of A171885 for n>=1 and is also mentioned in the example section of the same.
The number of bits of a(n) is equal to A129972(n).
Unary(n) = A105279(n-1).

			For n = 9, first part is "1110" and the second part is "001". So, a(9) = 1110001.

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. A105279 (unary code for n), A129972, A171885.

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, it is stored in 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 unary and the second part is stored in binary using floor(log_2 n) bits. Now the first and the second parts are concatenated to give the answer.

A010097 Prefix (or Levenshtein) codes for natural numbers.

Original entry on oeis.org

0, 2, 12, 13, 112, 113, 114, 115, 232, 233, 234, 235, 236, 237, 238, 239, 3840, 3841, 3842, 3843, 3844, 3845, 3846, 3847, 3848, 3849, 3850, 3851, 3852, 3853, 3854, 3855, 7712, 7713, 7714, 7715

Offset: 0

Leonid Broukhis

nonn

D. E. Knuth, "Supernatural Numbers", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 310-325.
D. E. Knuth, Selected Papers on Fun and Games, CSLI, 2011.
R. E. Krichevsky, Szhatie i poisk informatsii (Compressing and searching for information), Moscow, 1988, ISBN 5-256-00325-9.

Matthew House, Table of n, a(n) for n = 0..10000
Robert Munafo, Alternative Number Formats, section on "Lexicographic Strings".
Wikipedia, Levenshtein coding

Knuth articles also give A000918 and A171885.

apply( {A010097(n)=if(n, n+2^(n=exponent(n))*((n=A010097(n))+2<M. F. Hasler, Oct 24 2024

def encode(n):
    if n == 0: return "0"
    c, C = "", 1
    while n > 0:
        b = bin(n)[3:]
        c = b + c
        if (m := len(b)) > 0: C += 1
        n = m
    c = "1" * C + "0" + c
    return c
a = lambda n: int(encode(n),2) # Darío Clavijo, Aug 23 2024

The code for n is found as follows: from right to left, the truncated (without the leading 1) binary representations of n, floor(log_2(n)), floor(log_2(floor(log_2(n)))), etc., are written as long as they consist of at least one bit; then we write a 0 followed by log*(n) 1's.

Offset corrected by Matthew House, Aug 15 2016

A281223 The decimal representation of the Elias omega code for n (A281193(n)).

Original entry on oeis.org

0, 4, 6, 40, 42, 44, 46, 112, 114, 116, 118, 120, 122, 124, 126, 1312, 1314, 1316, 1318, 1320, 1322, 1324, 1326, 1328, 1330, 1332, 1334, 1336, 1338, 1340, 1342, 2752, 2754, 2756, 2758, 2760, 2762, 2764, 2766, 2768, 2770, 2772, 2774, 2776, 2778, 2780, 2782, 2784, 2786, 2788, 2790

Offset: 1

Indranil Ghosh, Jan 18 2017

			For n = 6, the Elias omega code for 6 is '101100' which is 44 in decimal. So, a(6) = 44.

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. A281193, A171885 (Decimal representation of the Elias gamma code for n).

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",2)

A281225 The decimal representation of the Elias delta code for n (A281150(n)).

Original entry on oeis.org

1, 8, 9, 20, 21, 22, 23, 192, 193, 194, 195, 196, 197, 198, 199, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857

Offset: 1

Indranil Ghosh, Jan 18 2017

			For n = 6, The Elias delta code for 6 is "10110" which is 22 in decimal. So, a(6) = 22.

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. A171885 (Decimal representation of the Elias gamma code for n), A281150, A281223.

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)

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.

cp's OEIS Frontend

A281149 Elias gamma code (EGC) for n.

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A010097 Prefix (or Levenshtein) codes for natural numbers.

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A281223 The decimal representation of the Elias omega code for n (A281193(n)).

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A281225 The decimal representation of the Elias delta code for n (A281150(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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