Grant Garcia - cp's OEIS frontend

A188215 Starting with an empty list, n is inserted after the a(n)th element such that the binary representations of the list's elements are always sorted lexicographically.

0, 1, 2, 3, 3, 4, 6, 7, 4, 5, 7, 8, 11, 12, 14, 15, 5, 6, 8, 9, 12, 13, 15, 16, 20, 21, 23, 24, 27, 28, 30, 31, 6, 7, 9, 10, 13, 14, 16, 17, 21, 22, 24, 25, 28, 29, 31, 32, 37, 38, 40, 41, 44, 45, 47, 48, 52, 53, 55

Offset: 0

Grant Garcia, Mar 24 2011

The last occurrence of any positive n in this sequence is a(2^(n - 1)).
As the list in question expands, its initial terms converge toward A131577.
The last item of the list is always zero or an element of A075427.

			For example, an a(n) of 3 means that n should be inserted after the 3rd element of the list to keep the elements lexicographically ordered.
[] (Initial empty list)
[0] (Zero inserted at the beginning: a(0) = 0)
[0, 1] (One inserted after element 1: a(1) = 1)
[0, 1, 10] (Two inserted after element 2: a(2) = 2)
[0, 1, 10, 11] (Three inserted after element 3: a(3) = 3)
[0, 1, 10, 100, 11] (Four inserted after element 3: a(4) = 3)

T. D. Noe, Table of n, a(n) for n = 0..1023

Cf. A264596.

lst = {}; Table[s = IntegerString[n, 2]; lst = Sort[Append[lst, s]]; Position[lst, s][[1, 1]] - 1, {n, 0, 63}] (* T. D. Noe, Apr 19 2011 *)

l = []
for i in range(17):
    b = bin(i)[2:]
    l.append(b)
    l.sort()
    print(l.index(b))

a(2^n + b) = n + b + 1 for b = 0 or 1.

a(2^n - b) = 2^n - b for b = 1 or 2.

Program added by Grant Garcia, Mar 30 2011

Edited by Grant Garcia, Apr 13 2011

A180625 a(1) = 2; a(n) is twice the previous term if it is prime, otherwise the previous term minus its lowest prime factor plus one.

Original entry on oeis.org

2, 4, 3, 6, 5, 10, 9, 7, 14, 13, 26, 25, 21, 19, 38, 37, 74, 73, 146, 145, 141, 139, 278, 277, 554, 553, 547, 1094, 1093, 2186, 2185, 2181, 2179, 4358, 4357, 8714, 8713, 17426, 17425, 17421, 17419, 34838, 34837, 34827

Offset: 1

Grant Garcia, Jan 21 2011

nonn

			2 is prime; 2 * 2 = 4. 4 is composite; 4 - lpf(4) + 1 = 4 - 2 + 1 = 3. 3 is prime; 3 * 2 = 6. 6 is composite; 6 - lpf(6) + 1 = 6 - 2 + 1 = 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020639.

Join[{s=2}, Table[If[PrimeQ[s], s=2s, s=s-FactorInteger[s][[1,1]]+1]; s, {43}]]
NestList[If[PrimeQ[#],2#,#-FactorInteger[#][[1,1]]+1]&,2,50] (* Harvey P. Dale, May 02 2012 *)

A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.

Original entry on oeis.org

3, 1, 4, 1, 6, 4, 0, 7, 8, 6, 4, 9, 9, 8, 7, 3, 8, 1, 7, 8, 4, 5, 5, 0, 4, 2, 0, 1, 2, 3, 8, 7, 6, 5, 7, 4, 1, 2, 6, 4, 3, 7, 1, 0, 1, 5, 7, 6, 6, 9, 1, 5, 4, 3, 4, 5, 6, 2, 5, 3, 8, 3, 4, 7, 2, 4, 6, 3, 1, 2, 5, 5, 5, 3, 8, 2, 6, 8, 2, 9, 3, 9, 6, 4, 8, 6, 4, 8, 6, 4, 5, 0, 2, 7, 2, 6, 9, 3, 6, 4, 9, 8, 1, 7, 0, 4, 9, 0, 5, 6, 9, 0, 4, 6

Offset: 1

Grant Garcia, Jan 16 2011

This is an approximation to Pi.

6*(phi+1)/5 is not equal to Pi, although some have claimed this (see Dudley). - Kellen Myers, Oct 04 2013

			3.141640786499873817845504201238765741264371015766915434562538347246312555382...

Underwood Dudley, Mathematical Cranks, MAA 1992, pp. 247, 292.
Alfred S. Posamentier and Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, New York, Prometheus Books, 2007, p. 119.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hung Viet Chu, Square the Circle in One Minute, arXiv:1908.01202 [math.GM], 2019.
Futility Closet, A Surprise Visitor

Cf. A022088, A022089, A001622, A000796.

(3/10)*(1 + Sqrt(5))^2; // G. C. Greubel, Jan 17 2018

RealDigits[(6/5)GoldenRatio^2, 10, 100][[1]] (* Alonso del Arte, Apr 09 2012 *)

3*(3+sqrt(5))/5 \\ Charles R Greathouse IV, Sep 13 2013

Limit of A022089(n+2)/A022088(n) as n approaches infinity.
6*(phi + 1)/5 = 6*phi^2/5 = 3(3 + sqrt(5))/5 = 9/5 + sqrt(9/5). - Charles R Greathouse IV, Sep 13 2013
Equals 24/(5-sqrt(5))^2. - Joost Gielen, Sep 20 2013

A179997 Where records occur in A179329.

Original entry on oeis.org

2, 4, 6, 9, 15, 25, 48, 91, 169, 336, 669, 1335, 2665, 5317, 10573, 21037, 42072, 84141, 168279, 336553, 673099, 1345639, 2691276, 5382549, 10765095, 21528959, 43057916, 86115821, 172231631, 344463251, 688926491, 1377847349, 2755694696, 5511375473, 11022750944

Offset: 1

Grant Garcia, Jan 13 2011

nonn

a(n + 1) / a(n) converges to 2.

Cf. A179329, A177980, A130904, A020639, A061228.

Corrected and extended by Grant Garcia, Jan 19 2011

a(30)-a(35) from Amiram Eldar, Oct 25 2024

A179329 Number of iterations of (n + lpf(n)) / 2 required to reach a prime, where lpf equals the least prime factor.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 3, 3, 0, 1, 0, 2, 4, 4, 0, 4, 0, 1, 2, 2, 0, 1, 5, 3, 5, 5, 0, 5, 0, 1, 5, 5, 2, 1, 0, 2, 3, 3, 0, 3, 0, 1, 2, 2, 0, 6, 6, 4, 6, 6, 0, 6, 6, 1, 6, 6, 0, 1, 0, 2, 6, 6, 3, 6, 0, 3, 2, 2, 0, 1, 0, 3, 4, 4, 4, 4, 0, 1, 4, 4, 0, 1, 3, 2, 3, 3, 0, 3, 7, 1, 7, 7, 5, 7, 0, 5, 7, 7, 0, 7, 0, 1, 7, 7

Offset: 2

Grant Garcia, Jan 12 2011

nonn

Zeros indicate prime indices.

			a(15) gives (15 + 3) / 2 = 9, (9 + 3) / 2 = 6, (6 + 2) / 2 = 4, (4 + 2) / 2 = 3. Four iterations were required to reach a prime, so a(15) = 4.

Cf. A177980, A020639, A061228.

f[n_] := Length@ NestWhileList[(# + FactorInteger[#][[1, 1]])/2 &, n, ! PrimeQ@ # &] - 1; Array[f, 105, 2]

A177980 Iterate (n + lpf(n)) / 2 until a prime is reached, where lpf equals the least prime factor. a(n) is that terminating prime.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 5, 3, 3, 11, 7, 13, 5, 3, 3, 17, 3, 19, 11, 7, 7, 23, 13, 3, 5, 3, 3, 29, 3, 31, 17, 3, 3, 11, 19, 37, 11, 7, 7, 41, 7, 43, 23, 13, 13, 47, 3, 3, 5, 3, 3, 53, 3, 3, 29, 3, 3, 59, 31, 61, 17, 3, 3, 11, 3, 67, 11, 19, 19, 71, 37, 73, 11

Offset: 2

Grant Garcia, Dec 16 2010

nonn

The function (n + lpf(n)) / 2 reduces the input according to its lowest prime factor if it is composite or simply returns the input if it is prime.

Sequence contains only prime numbers (and every prime number).

			7 is prime, so (7 + lpf(7)) / 2 = (7 + 7) / 2 = 7.
15 is composite: (15 + 3) / 2 = 9, (9 + 3) / 2 = 6, (6 + 2) / 2 = 4, (4 + 2) / 2 = 3.

Dumitru Damian, Table of n, a(n) for n = 2..30000

Cf. A020639, A061228.

g[n_] := (n + FactorInteger[n][[1, 1]])/2; f[n_] := Last@ NestWhileList[g, n, !PrimeQ@ # &]; Array[f, 73, 2]

from sympy import factorint, isprime
def a177980(n):
    while True:
        if isprime(n): return n
        else: n=int((n+A020639(n))/2)
[a177980(n) for n in range(2, 160)] # Dumitru Damian, Dec 15 2021

A177869 Integers divisible by their number of digits in binary.

Original entry on oeis.org

1, 2, 6, 8, 12, 20, 25, 30, 36, 42, 48, 54, 60, 70, 77, 84, 91, 98, 105, 112, 119, 126, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405

Offset: 1

Grant Garcia, Dec 13 2010

			105 is 1101001 in base 2 (length of 7); 105 / 7 is 15.

Daniel Arribas, Table of n, a(n) for n = 1..1000

Base 10 equivalent is A098952.

base_weight b g n | n == 0 = 0 | otherwise = (base_weight b g (n `div` b)) + (g $ n `mod` b)
interesting b g = filter f [1..] where f n = n `mod` (base_weight b g n) == 0
bin_interesting g = interesting 2 g
weights l n | (n >=0) && ((length l) > fromInteger n) = l !! fromInteger n | otherwise = 0
cnst = weights [1, 1]
let sequence = bin_interesting cnst -- Victor S. Miller, Oct 17 2011

Select[Range[410], IntegerQ[#/Length[IntegerDigits[#, 2]]] &] (* Alonso del Arte, Dec 13 2010 *)

for(d=1, 9, forstep(n=(2^(d-1)+d-1)\d*d, 2^d-1, d, print1(n", "))) \\ Charles R Greathouse IV, Oct 17 2011

import math
for n in range(1, 1000):
    if not n % int(math.log(n, 2) + 1): print(n) # Garcia

A181709 Indices of primes in A007310.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 23, 24, 25, 27, 28, 30, 33, 34, 35, 36, 37, 38, 43, 44, 46, 47, 50, 51, 53, 55, 56, 58, 60, 61, 64, 65, 66, 67, 71, 75, 76, 77, 78, 80, 81, 84, 86, 88, 90, 91, 93, 94, 95, 98, 103, 104, 105, 106, 111, 113, 116, 117, 118

Offset: 1

Grant Garcia, Nov 07 2010

All primes but 2 and 3 are present in A007310, making this sequence an efficient method for storing large quantities of primes. To unpack this sequence into primes, use the formula (6n + (-1)^n - 3) / 2.

Indices 1 and 9 (1 and 25) are the smallest nonprimes.

			A007310(2), 5, is the first prime of the sequence.
A007310(50), 149, is also a prime, hence 50 is included.

Grant Garcia, Table of n, a(n) for n=1..10000

Cf. A007310, A000040.

Floor[Prime[Range[3,80]]/3]+1 (* Harvey P. Dale, Sep 12 2019 *)

from sympy import isprime
out = ""
for n,p in enumerate(isprime((6*n+(-1)**n-3)//2) for n in range(1,1000)):
    out+=["","%s "%str(n+1)][p]
for n,p in enumerate(out.rstrip(" ").split(" ")): print(n+1,p)

a(n) = floor(prime(n+2)/3)+1 = A144769(n+2)+1. - Gary Detlefs, Dec 11 2011

a(n) ~ n*log(n)/3. - Ilya Gutkovskiy, Jul 18 2016

A175941 Number of n-step self-avoiding walks on a line, where step X skips X - 1 spaces.

Original entry on oeis.org

1, 2, 4, 6, 10, 18, 30, 50, 78, 130, 210, 350, 586, 954, 1606, 2588, 4234, 6944, 11342, 18948, 31450, 52206, 85662, 141680, 233040, 385428, 644910, 1072074, 1783342, 2953094, 4897922, 8157096, 13571014, 22552212, 37486916, 62325564, 103508754, 172765524, 287428656, 479052200, 798944976

Offset: 0

Grant Garcia, Oct 25 2010

			a(0) = 1 because every walk must start at a specific point on the line.
   --0--
a(1) = 2 because there are two ways to start, moving one space forward or backward.
   --01-
   -10--
a(2) = 4 because there are two ways to continue each of the walks described in a(1).
   ---01-2
   --201--
   --102--
   2-10---
a(3) = 6, not 8, because two of the possible four continuations are not self-avoiding.
   ------01-2--3
   -----2013----
   --3--201-----
   -----102--3--
   ----3102-----
   3--2-10------

Cf. A321535.

b:= proc(n, s) option remember; `if`(n=0, 1, add(`if`(j in s, 0,
      b(n-1, {map(x-> x-j, s)[], 0})), j=(k->[-k, k])(nops(s))))
    end:
a:= n-> b(n, {0}):
seq(a(n), n=0..24);  # Alois P. Heinz, May 19 2021

b[n_, s_] := b[n, s] = If[n == 0, 1, Sum[If[MemberQ[s, j], 0, b[n - 1, Union[Append[s - j, 0]]]], {j, {-Length[s], Length[s]}}]];
a[n_] := b[n, {0}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 09 2021, after Alois P. Heinz *)

a(n)={local(f=vectorsmall(n*(n+1)+1)); my(recurse(p, k)=if(!f[p], if(k==n, 1, f[p]=1; k++; my(z=self()(p+k, k) + self()(p-k, k)); f[p]=0; z))); recurse((#f+1)/2, 0)} \\ Andrew Howroyd, Nov 12 2018

x = 1
n = 0
runs = [[n]]
while x < 30:
    print(x - 1, n)
    runs2 = []
    n = 0
    while runs:
        for pmx in (x, -x):
            if runs[0][ -1] + pmx not in runs[0]:
                runs2.append(runs[0] + [runs[0][ -1] + pmx])
                n += 1
        del runs[0]
    runs = runs2
    x += 1

a(n) <= 2*a(n-1) for n > 0. - Andrew Howroyd, Nov 12 2018

a(29)-a(40) from Andrew Howroyd, Nov 12 2018

A175924 Smallest power of 2 with n repeated digits.

Original entry on oeis.org

1, 65536, 16777216, 2199023255552, 1684996666696914987166688442938726917102321526408785780068975640576

Offset: 1

Grant Garcia, Oct 18 2010, Oct 20 2010

The subsequent terms are too large to display.
a(6) and a(7), 2^971 and 2^972, respectively, both of which have 293 digits; a(8), 2^8554, has 2576 digits. a(9), 2^42485, has 12790 digits.
Corresponding exponents of 2 are 0, 16, 24, 41, 220, 971, 972, 8554, 42485, 42486, 271979. [Zak Seidov, Oct 19 2010]

			a(1) is 1 because it is the first power of 2; all integers have at least one digit.
a(2) is 65536 because it is the first power of 2 with two of the same digit in a row.
a(3) is 16777216 because it is the first power of 2 with three of the same digit in a row.

Wikipedia, Power of 2

Subsequence of A000079 (powers of 2).

Cf. A045875.

f[n_] := Block[{k = 0}, While[ !MemberQ[Length /@ Split@ IntegerDigits[2^k], n], k++ ]; 2^k]; Table[f[n], {n, 5}] (* Robert G. Wilson v, Oct 21 2010 *)

import math
for N in range(1, 10):
    repdigits = 1
    n = 0
    while repdigits < N:
        n += 1
        s = str(2 ** n)
        prev = ""
        repdigits = maxrepdigits = 1
        for d in s:
            if d == prev: repdigits += 1
            else:
                maxrepdigits = max(maxrepdigits, repdigits)
                repdigits = 1
            prev = d
        repdigits = max(maxrepdigits, repdigits)
    print(N, 2 ** n)

cp's OEIS Frontend

User: Grant Garcia

A188215 Starting with an empty list, n is inserted after the a(n)th element such that the binary representations of the list's elements are always sorted lexicographically.

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A180625 a(1) = 2; a(n) is twice the previous term if it is prime, otherwise the previous term minus its lowest prime factor plus one.

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Mathematica

A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.

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Magma

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A179997 Where records occur in A179329.

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A179329 Number of iterations of (n + lpf(n)) / 2 required to reach a prime, where lpf equals the least prime factor.

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Mathematica

A177980 Iterate (n + lpf(n)) / 2 until a prime is reached, where lpf equals the least prime factor. a(n) is that terminating prime.

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A177869 Integers divisible by their number of digits in binary.

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Haskell

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PARI

Python

A181709 Indices of primes in A007310.

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