cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A138148 Cyclops numbers with binary digits only.

Original entry on oeis.org

0, 101, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111
Offset: 0

Views

Author

Omar E. Pol, Mar 18 2008

Keywords

Comments

All members are palindromes A002113. The first five members are mentioned in A129868.
Also, binary representation of A129868.
a(A090748(n)) is equal to A138831(n), the n-th perfect number minus 1, written in base 2.
Except for the first term (replace 0 with 1) the binary representation of the n-th iteration of the elementary cellular automaton, Rule 219 starting with a single ON (black) cell. - Robert Price, Feb 21 2016
a(1) = 101 is only prime number in this sequence since a(n) = (10^(n+1)+1)*(10^n-1)/9. - Altug Alkan, May 11 2016

Examples

			n ........ a(n) .... A129868(n): value of a(n) read in base 2.
0 ......... 0 ......... 0
1 ........ 101 ........ 5
2 ....... 11011 ....... 27
3 ...... 1110111 ...... 119
4 ..... 111101111 ..... 495
5 .... 11111011111 .... 2015
6 ... 1111110111111 ... 8127

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cyclops numbers: A134808. Cf. A002113, A129868.
Cf. A000396, A090748, A135627, A138831.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).

Programs

Formula

From Colin Barker, Feb 21 2013: (Start)
a(n) = (-1-9*10^n+10^(1+2*n))/9.
G.f.: x*(200*x-101) / ((x-1)*(10*x-1)*(100*x-1)). (End)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2. - Wesley Ivan Hurt, Dec 08 2015
a(n) = A000533(n+1)*A002275(n). - Altug Alkan, May 12 2016
E.g.f.: (-1 - 9*exp(9*x) + 10*exp(99*x))*exp(x)/9. - Ilya Gutkovskiy, May 12 2016
a(n) = A002275(2n+1) - A011557(n). - M. F. Hasler, Feb 08 2020

Extensions

More terms from Omar E. Pol, Feb 09 2020

A089633 Numbers having no more than one 0 in their binary representation.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 191, 223, 239, 247, 251, 253, 254, 255, 383, 447, 479, 495, 503, 507, 509, 510, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1023
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 01 2004

Keywords

Comments

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009
Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012
Numbers of the form 2^t - 2^k - 1, 0 <= k < t.
n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018
Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

Examples

			From _Tilman Piesk_, May 09 2012: (Start)
This may also be viewed as a triangle:             In binary:
                  0                                         0
               1     2                                 01       10
             3    5    6                          011      101      110
           7   11   13   14                  0111     1011     1101     1110
        15   23   27   29   30          01111    10111    11011    11101    11110
      31  47   55   59   61   62
   63   95  111  119  123  125  126
Left three diagonals are A000225,  A055010, A086224. Right diagonal is A000918. Central column is A129868. Numbers in row n (counted from 0) have n binary 1s. (End)
From _Gus Wiseman_, May 24 2024: (Start)
The terms together with their binary expansions and binary indices begin:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
  11:   1011 ~ {1,2,4}
  13:   1101 ~ {1,3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  23:  10111 ~ {1,2,3,5}
  27:  11011 ~ {1,2,4,5}
  29:  11101 ~ {1,3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  47: 101111 ~ {1,2,3,4,6}
  55: 110111 ~ {1,2,3,5,6}
  59: 111011 ~ {1,2,4,5,6}
  61: 111101 ~ {1,3,4,5,6}
  62: 111110 ~ {2,3,4,5,6}
(End)

Links

Crossrefs

Cf. A000120, A007088, A011371, A014132, A023416, A023532, A029931.
Cf. A181741 (primes), union of A081118 and A000918, apart from initial -1.
Cf. A065442, A160502, A265705.
For least binary index (instead of rank) we have A001511.
Applying A019565 (Heinz number of binary indices) gives A077011.
For greatest binary index we have A029837 or A070939, opposite A070940.
Row minima of A118462 (binary ranks of strict partitions).
For sum instead of minimum we have A372888, non-strict A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A277905 groups all positive integers by binary rank of prime indices.
Cf. A000041, A005940, A018819, A147655, A225620, A246868.

Programs

  • Haskell
    a089633 n = a089633_list !! (n-1)
a089633_list = [2 ^ t - 2 ^ k - 1 | t <- [1..], k <- [t-1,t-2..0]]
-- Reinhard Zumkeller, Feb 23 2012
  • Maple
    seq(seq(2^a-1-2^b,b=a-1..0,-1),a=1..11); # Robert Israel, Dec 14 2018
  • Mathematica
    fQ[n_] := DigitCount[n, 2, 0] < 2; Select[ Range[0, 2^10], fQ] (* Robert G. Wilson v, Aug 02 2012 *)
  • PARI
    {insq(n) = local(dd, hf, v); v=binary(n);hf=length(v);dd=sum(i=1,hf,v[i]);if(dd<=hf-2,-1,1)}
{for(w=0,1536,if(insq(w)>=0,print1(w,", ")))}
\\ Douglas Latimer, May 07 2013
  • PARI
    isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018
  • Python
    from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<A089633_list = list(islice(A089633_gen(),30)) # Chai Wah Wu, Feb 10 2023
  • Python
    from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

Formula

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;
A000120(a(u)) <= A000120(a(v)) for uA000120(a(n)) = A003056(n).
a(0)=0, n>0: a(n+1) = Min{m>n: BinOnes(a(n))<=BinOnes(m)} with BinOnes=A000120.
If m = floor((sqrt(8*n+1) - 1) / 2), then a(n) = 2^(m+1) - 2^(m*(m+3)/2 - n) - 1. - Carl R. White, Feb 10 2009
A029931(a(n)) = n and A029931(m) != n for m < a(n). - Reinhard Zumkeller, Feb 28 2014
A265705(a(n),k) = A265705(a(n),a(n)-k), k = 0 .. a(n). - Reinhard Zumkeller, Dec 15 2015
a(A014132(n)-1) = 2*a(n-1)+1 for n >= 1. - Robert Israel, Dec 14 2018
Sum_{n>=1} 1/a(n) = A065442 + A160502 = 3.069285887459... . - Amiram Eldar, Jan 09 2024
A019565(a(n)) = A077011(n). - Gus Wiseman, May 24 2024

A285332 a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 8, 15, 12, 14, 27, 10, 25, 7, 16, 210, 45, 35, 18, 105, 28, 462, 81, 21, 20, 154, 125, 30, 49, 11, 32, 10659, 420, 910, 75, 78, 175, 33, 24, 3094, 315, 385, 56, 780045, 924, 374, 243, 110, 63, 55, 40, 4389, 308, 170170, 625, 1155, 60, 286, 343, 42, 121, 13, 64, 54230826, 31977, 28405, 630, 1330665, 1820, 714
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:
1
|
...................2...................
3 4
6......../ \........9 5......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
15 12 14 27 10 25 7 16
210 45 35 18 105 28 462 81 21 20 154 125 30 49 11 32
etc.
Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.
As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.
Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.
However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

Links

Crossrefs

Inverse: A285331.
Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333, A286542, A286543.
Compare also to permutation A285112 and array A285321.

Programs

  • Mathematica
    Block[{a = {1, 2}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[a[[i/2 + 1]], 2], If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &[a[[(i - 1)/2 + 1]] ] ]], {i, 2, 70}]; a] (* Michael De Vlieger, Mar 12 2021 *)
  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
for(n=0, 4095, write("b285332.txt", n, " ", A285332(n)));
  • Python
    from operator import mul
from sympy import prime, primefactors
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
def a(n):
    if n<2: return n + 1
    if n%2==0: return a019565(a(n//2))
    else: return a065642(a((n - 1)//2))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017
  • Scheme
    ;; With memoization-macro definec.
(definec (A285332 n) (cond ((<= n 1) (+ n 1)) ((even? n) (A019565 (A285332 (/ n 2)))) (else (A065642 (A285332 (/ (- n 1) 2))))))

Formula

a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).
For n >= 0, a(2^n) = A109162(2+n). [The left edge of the tree.]
For n >= 0, a(A000225(n)) = A000079(n). [Powers of 2 occur at the right edge of the tree.]
For n >= 2, a(A000918(n)) = A000040(n). [And the next vertices inwards contain primes.]
For n >= 2, a(A036563(1+n)) = A001248(n). [Whose right children are their squares.]
For n >= 0, a(A055010(n)) = A000244(n). [Powers of 3 are at the rightmost edge of the left subtree.]
For n >= 2, a(A129868(n-1)) = A062457(n).
A048675(a(n)) = A285333(n).
A046523(a(n)) = A286542(n).

A190620 Odd numbers with a single zero in their binary expansion.

Original entry on oeis.org

5, 11, 13, 23, 27, 29, 47, 55, 59, 61, 95, 111, 119, 123, 125, 191, 223, 239, 247, 251, 253, 383, 447, 479, 495, 503, 507, 509, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 3071, 3583, 3839, 3967, 4031
Offset: 1

Views

Author

Reinhard Zumkeller, May 14 2011

Keywords

Comments

Odd numbers such that the binary weight is one less than the number of significant digits. Except for the initial 0, A129868 is a subsequence of this sequence. - Alonso del Arte, May 14 2011
From Bernard Schott, Oct 20 2022: (Start)
A036563 \ {-2, -1, 1} is a subsequence, since for m >= 3, A036563(m) = 2^m - 3 has 11..1101 with (m-2) starting 1's for binary expansion.
A083329 \ {1, 2} is a subsequence, since for m >= 2, A083329(m) = 3*2^(m-1) - 1 has 1011..11 with (m-1) trailing 1's for binary expansion.
A129868 \ {0} is a subsequence, since for m >= 1, A129868(m) = 2*4^m - 2^m - 1 is a binary cyclops number that has 11..11011..11 with m starting 1's and m trailing 1's for binary expansion.
The 0-bit position in binary expansion of a(n) is at rank A004736(n) + 1 from the right.
For k >= 2, there are (k-1) terms between 2^k and 2^(k+1), or equivalently (k-1) terms with (k+1) bits.
{2*a(n), n>0} form a subsequence of A353654 (numbers with one trailing 0 bit and one other 0 bit). (End)

Links

Crossrefs

Cf. A007088, A023416, A190619, A353654.
A036563 \ {-2, -1, 1}, A083329 \ {1, 2}, A129868 are subsequences.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A357773 (k=2).

Programs

  • Haskell
    import Data.List (elemIndices)
a190620 n = a190620_list !! (n-1)
a190620_list = filter odd $ elemIndices 1 a023416_list
-- A more efficient version, inspired by the Maple program in A190619:
a190620_list' = g 8 2 where
   g m 2 = (m - 3) : g (2*m) (m `div` 2)
   g m k = (m - k - 1) : g m (k `div` 2)
  • Maple
    isA := proc(n) convert(n, base, 2): %[1] = nops(%) - add(%) end:
select(isA, [$1..4031]); # Peter Luschny, Oct 27 2022
# Alternatively, using a formula of Bernard Schott and A123578:
A190620 := proc(n) A123578(n); 4*2^% - 2^(1 - n + (% + %^2)/2) - 1 end:
seq(A190620(n), n = 1..50); # Peter Luschny, Oct 28 2022
  • Mathematica
    Select[Range[1,5001,2],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Jul 12 2018 *)
  • Python
    from itertools import count, islice
def agen():
    for d in count(3):
        b = 1 << d
        for i in range(2, d):
            yield b - (b >> i) - 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2022
  • Python
    from math import isqrt, comb
def A190620(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1)+1)-(1<Chai Wah Wu, Dec 18 2024

Formula

A190619(n) = A007088(a(n));
A023416(a(n)) = 1.
From Bernard Schott, Oct 21 2022: (Start)
a((n-1)*(n-2)/2 - (i-1)) = 2^n - 2^i - 1 for n >= 3 and 1 <= i <= n-2 (after Robert Israel in A357773).
a(n) = A000225(A002024(n)+2) - A000079(A004736(n)).
a(n) = 4*2^k(n) - 2^(1 - n + (k(n) + k(n)^2)/2) - 1, where k is the Kruskal-Macaulay function A123578.
A070939(a(n)) = A002024(n) + 2. (End)

A281482 a(n) = 2^(n + 1) * (2^n + 1) - 1.

Original entry on oeis.org

3, 11, 39, 143, 543, 2111, 8319, 33023, 131583, 525311, 2099199, 8392703, 33562623, 134234111, 536903679, 2147549183, 8590065663, 34360000511, 137439477759, 549756862463, 2199025352703, 8796097216511, 35184380477439, 140737505132543, 562949986975743
Offset: 0

Views

Author

Jaroslav Krizek, Jan 22 2017

Keywords

Links

Crossrefs

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281481 (2^(n - 1) * (2^n + 1) + 1).

Programs

  • Magma
    [2^(n + 1) * (2^n + 1) - 1: n in [0..200]];
  • PARI
    Vec((3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A249544 Array read by antidiagonals: T(m,n) read in binary is a palindrome with m runs of n ones separated by single zeros.

Original entry on oeis.org

1, 3, 5, 7, 27, 21, 15, 119, 219, 85, 31, 495, 1911, 1755, 341, 63, 2015, 15855, 30583, 14043, 1365, 127, 8127, 128991, 507375, 489335, 112347, 5461, 255, 32639, 1040319, 8255455, 16236015, 7829367, 898779, 21845, 511, 130815, 8355711
Offset: 1

Views

Author

Tilman Piesk, Oct 31 2014

Keywords

Comments

The entries in this array are all in A194602, and therefore can be interpreted as integer partitions: T(m,n) is the integer partition with m times the addend n+1, and no other non-one addends. The array A249543 contains the corresponding indices of A194602.

Examples

			Array starts:                                          Binary:
  n    1      2       3         4          5
m
1      1      3       7        15         31               1        11          111
2      5     27     119       495       2015             101     11011      1110111
3     21    219    1911     15855     128991           10101  11011011  11101110111
4     85   1755   30583    507375    8255455
5    341  14043  489335  16236015  528349151

Links

Crossrefs

Cf. A249543, A194602; Rows: A000225, A129868; Columns: A002450, A083713.

Programs

  • PHP
    A249544($m, $n) {
// a                  b            c
// ( 2^(n+1)^m -1 ) * ( 2^n -1 ) / ( 2^(n+1) -1 )
$a = gmp_sub( gmp_pow( gmp_pow(2,$n+1), $m ), 1 );
$b = gmp_sub( gmp_pow(2,$n), 1 );
$c = gmp_sub( gmp_pow(2,$n+1), 1 );
$return = gmp_div_q( gmp_mul($a,$b), $c );
return gmp_strval($return);
}

Formula

T(m,n) = ( 2^(n+1)^m -1 ) * ( 2^n -1 ) / ( 2^(n+1) -1 ).

A267685 Decimal representation of the n-th iteration of the "Rule 203" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527, 140737479966719, 562949936644095
Offset: 0

Views

Author

Robert Price, Jan 19 2016

Keywords

Comments

Conjectures from Barker confirmed by later formulas. - Ray Chandler, Aug 09 2025

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A129868, A267683, A267684.

Programs

  • Mathematica
    rule=203; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)
  • Python
    print([1, 4]+[2*4**n - 2**n - 1 for n in range(2, 50)]) # Karl V. Keller, Jr., Jun 07 2022

Formula

From Colin Barker, Jan 19 2016 and Apr 17 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.
G.f.: (1-3*x+13*x^2-22*x^3+8*x^4) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = A129868(n) for n >= 2. - Georg Fischer, Mar 26 2019
a(n) = 2*4^n - 2^n - 1 for n > 1. - Karl V. Keller, Jr., Jun 07 2022

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 23, 111, 479, 1983, 8063, 32511, 130559, 523263, 2095103, 8384511, 33546239, 134201343, 536838143, 2147418111, 8589803519, 34359476223, 137438429183, 549754765311, 2199021158399, 8796088827903, 35184363700223, 140737471578111, 562949919866879
Offset: 0

Views

Author

Robert Price, Jan 20 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267814.
Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - Jaroslav Krizek, Jan 22 2017

Programs

  • Mathematica
    rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 22 2016 and Apr 16 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - Jaroslav Krizek, Jan 22 2017

A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

Original entry on oeis.org

2, 4, 11, 37, 137, 529, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529
Offset: 0

Views

Author

Jaroslav Krizek, Jan 22 2017

Keywords

Links

Crossrefs

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281482 (2^(n + 1) * (2^n + 1) - 1).
Cf. A278930.

Programs

  • Magma
    [2^(n - 1) * (2^n + 1) + 1: n in [0..200]];
  • PARI
    Vec((2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
a(n) = A278930(n - 2) for n >= 7. - Georg Fischer, Mar 26 2019

A369317 a(n) = A091255(n, n + 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 3, 1, 7, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1
Offset: 1

Views

Author

Rémy Sigrist, Jan 19 2024

Keywords

Comments

Two consecutive integers are always coprime, however the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive integers are not necessarily coprime, hence this sequence.

Examples

			The first terms, alongside the correspond GF(2)[X]-polynomials, are:
  n   a(n)  P(n)               P(n+1)             gcd(P(n), P(n+1))
  --  ----  -----------------  -----------------  -----------------
   1     1  1                  X                  1
   2     1  X                  X + 1              1
   3     1  X + 1              X^2                1
   4     1  X^2                X^2 + 1            1
   5     3  X^2 + 1            X^2 + X            X + 1
   6     1  X^2 + X            X^2 + X + 1        1
   7     1  X^2 + X + 1        X^3                1
   8     1  X^3                X^3 + 1            1
   9     3  X^3 + 1            X^3 + X            X + 1
  10     1  X^3 + X            X^3 + X + 1        1

Links

Crossrefs

Cf. A091255, A129868, A369277 (distinct values), A369318 (indices of values <> 1).

Programs

  • PARI
    a(n) = fromdigits(lift(Vec(gcd(Mod(1, 2) * Pol(binary(n)), Mod(1, 2) * Pol(binary(n+1))))), 2)

Formula

a(A129868(k)) = 2^(k+1) - 1 for any k > 0.
Showing 1-10 of 18 results. Next

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