A083420 -id:A083420 - cp's OEIS frontend

A303449 Denominator of (2n+1)/(2^(2n+1)-1).

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 299593, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311, 2251799813685247

Offset: 0

Altug Alkan, Apr 24 2018

If A160145(n) = 0, then a(n) = A083420(n).

Least values of k such that a(k) = A083420(k)/A036259(n) are 0, 10, 126, 77, 540, 73, 1242, 328, 1540, 489 for 1 <= n <= 10.

Cf. A005408, A036259, A083420, A160144 (numerators), A160145.

seq(denom((2*n+1)/(2^(2*n+1)-1)), n=0..25);

a(n) = denominator((2*n+1)/(2^(2*n+1)-1));

forstep(k=1, 1e2, 2, print1(denominator(k/(2^k-1)), ", "));

A331895 Positive numbers k such that the binary and negabinary representations of k and the negabinary representation of -k are all palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 65, 85, 127, 257, 273, 325, 341, 455, 511, 1025, 1105, 1285, 1365, 1799, 2047, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 7175, 7967, 8191, 16385, 16705, 17425, 17745, 20485, 20805, 21525, 21845, 28679, 29127, 31775, 32767, 65537, 65793

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			7 is a term since the binary representation of 7, 111, the negabinary representation of 7, 11011, and the negabinary representation of -7, 1001, are all palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A006995 and A331893.
Intersection of A331892 and A331894.
Cf. A039724, A331891.

binPalinQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := And @@(PalindromeQ @ negabin[#] & /@ {n, -n}); Select[Range[2^16], binPalinQ[#] && nbPalinQ[#] &]

A368275 Fibonacci zig-zag function.

Original entry on oeis.org

1, 1, 2, 3, 6, 4, 10, 6, 11, 10, 15, 11, 17, 15, 18, 17, 22, 18, 26, 22, 27, 26, 29, 27, 33, 29, 34, 33, 36, 34, 40, 36, 41, 40, 45, 41, 49, 45, 50, 49, 54, 50, 56, 54, 57, 56, 61, 57, 63, 61, 64, 63, 68, 64, 70, 68, 71, 70, 75, 71, 77, 75, 78, 77, 82, 78, 86

Offset: 0

Darío Clavijo, Dec 19 2023

nonn

The craft of coding, Weird recursive algorithms, zib(n).
Thomas Scheuerle, Plot of a(n) - (5/4)*n for n = 1..200.
Thomas Scheuerle, Plot of a(n) - (5/4)*n for n = 1..100000.

Cf. A000045, A083420, A016813.

function a = A368275( max_n )
     a = [1, 1, 2];
     for m = 4:max_n
         n = m-1;
         if mod(n,2) == 0
            a(m) = a(1 + (n/2)) + a(2 + (n/2)) + 1;
         else
            a(m) = a(1 + (n-1)/2) + a((n-1)/2) + 1;
         end
     end
end % Thomas Scheuerle, Dec 19 2023

a[0] = 1;a[1] = 1;a[2] = 2;a[n_Integer] := a[n] = Which[n == 0,1,n == 1,1,n == 2, 2,Mod[n, 2] == 1 && n >= 3, a[Quotient[(n-1),2]] + a[Quotient[(n-1),2]-1]+1,Mod[n,2] == 0 && n >= 4, a[Quotient[n,2]] + a[Quotient[n,2]+ 1]+1];result = Table[a[n],{n,0,66}] (* James C. McMahon, Dec 19 2023 *)

from functools import cache
@cache
def a(n):
  if n < 3: return [1,1,2][n]
  x, y = (n-1)>>1, n>>1
  if n & 1 == 1: return a(x) + a(x-1) + 1
  else: return a(y) + a(y+1) + 1
print([a(n) for n in range(0,67)])

a(0)=1 and a(1)=1 and a(2)=2.
a(2*n+1) = a(n) + a(n-1) + 1, for n>=1.
a(2*n) = a(n) + a(n+1) + 1, for n>=2.
From Thomas Scheuerle, Dec 19 2023: (Start)
a(4^n+1) = (5*4^n - 8)/4, for n > 1.
a(2*4^n-1) = (5*4^n - 8)/2, for n > 0. (End)

A232710 Binary numbers (written in decimal) such that the sum of digits mod 2 equals the product of digits mod 2.

Original entry on oeis.org

0, 1, 5, 6, 7, 9, 10, 12, 17, 18, 20, 23, 24, 27, 29, 30, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125

Offset: 0

Jon Perry, Nov 28 2013

			23 is in the sequence because 23 in binary is 10111. The sum of digits is 4 == 0 mod 2 and the products of digits is 0 == 0 mod 2.

Index entries for sequences related to binary expansion of n

for (i=0;i<1000;i++) {
s=i.toString(2).split("");
sl=s.length;
c=0;d=1;
for (j=0;j

is(n)=b=binary(n);sum(i=1,#b,b[i])%2==prod(i=1,#b,b[i])%2 \\ Ralf Stephan, Nov 30 2013

A083420 UNION (A001969 \ {3}). - Ralf Stephan, Nov 30 2013

A233868 Numbers that are not the sum of two evil numbers.

Original entry on oeis.org

1, 2, 4, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831

Offset: 1

Juri-Stepan Gerasimov, Dec 17 2013

2, 4 together with A083420. - R. J. Mathar, Dec 22 2013.

Also, numbers that are not the sum of two distinct odious numbers. - David Radcliffe, Jul 21 2016

Index entries for linear recurrences with constant coefficients, signature (5, -4).

Cf. A000069, A000668, A001969, A083420, A175611 (primes that are not the sum of two positive evil numbers).

From Chai Wah Wu, Aug 02 2020: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n > 5.
G.f.: x*(12*x^4 - 5*x^3 - 2*x^2 - 3*x + 1)/((x - 1)*(4*x - 1)). (End)

Terms corrected by R. J. Mathar, Dec 22 2013

A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.

Original entry on oeis.org

2, 1, 11, 7, 47, 31, 191, 127, 767, 511, 3071, 2047, 12287, 8191, 49151, 32767, 196607, 131071, 786431, 524287, 3145727, 2097151, 12582911, 8388607, 50331647, 33554431, 201326591, 134217727, 805306367, 536870911, 3221225471, 2147483647, 12884901887, 8589934591

Offset: 2

Vincenzo Librandi, May 07 2018

A198693 and A083420 interleaved. From 11 onwards, apparently A283651 and A290195 contain the same terms. - Bruno Berselli, May 07 2018

Olivier Rozier, Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding, arXiv:1805.00133 [math.DS], 2018-2025, see Definition 4.4 on p. 21.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

All terms belong to A052955 and A180516.

Cf. A083420, A198693, A283651, A290195.

[IsOdd(n) select 2^(n-2)-1 else 3*2^(n-2)-1: n in [2..40]];

I:=[2,1,11]; [n le 3 select I[n] else Self(n-1)+4*Self(n-2)-4*Self(n-3): n in [1..35]];

Table[If[OddQ[n], 2^(n - 2) - 1, 3 2^(n - 2) - 1], {n, 2, 80}]
LinearRecurrence[{1, 4, -4}, {2, 1, 11}, 30]

a(n) = if (n%2, 2^(n-2) - 1, 3*2^(n-2) - 1); \\ Michel Marcus, May 30 2018

a(n) = 2^(n-2) - 1 for odd n, otherwise a(n) = 3*2^(n-2) - 1, with n>1.
From Bruno Berselli, May 07 2018: (Start)
O.g.f.: x^2*(2 - x + 2*x^2)/((1 - x)*(1 - 2*x)*(1 + 2*x)).
E.g.f.: (1 + 2*x - 4*exp(x) + exp(-2*x) + 2*exp(2*x))/4.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
a(n) = (2 + (-1)^n)*2^(n-2) - 1. (End)

A308267 Numbers k such that k divides A048678(k).

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 31, 32, 56, 62, 64, 83, 112, 124, 127, 128, 166, 224, 248, 254, 256, 332, 397, 448, 496, 508, 511, 512, 664, 794, 891, 896, 992, 1016, 1022, 1024, 1163, 1328, 1588, 1782, 1792, 1984, 2032, 2044, 2047, 2048, 2326, 2656, 3176, 3441, 3564, 3584, 3968

Offset: 1

Dan Dart, May 17 2019

Apparently includes all powers of 2.

All numbers 2^(2k+1)-1 are in this sequence, and if n is in this sequence then so is 2n. - Charlie Neder, May 17 2019

			The first few terms of A048678 are 1,2,5,4,9,10,21,8. 2 is a multiple of 2, 5 isn't a multiple of 3, 4 is a multiple of 4, 9 isn't a multiple of 5, 10 isn't a multiple of 6, 21 is a multiple of 7, etc.

Cf. A048678, A083420 (subsequence).

bintodec :: [Integer] -> Integerbintodec = sum . zipWith (*) (iterate (*2) 1) . reverse
decomp :: (Integer, [Integer]) -> (Integer, [Integer])decomp (x, ys) = if even x then (x `div` 2, 0:ys) else (x - 1, 1:ys)
zeck :: Integer -> Integerzeck n = bintodec (1 : snd (last . takeWhile (\(x, _) -> x > 0) $ iterate decomp (n, [])))
output :: [Integer]output = filter (\x -> 0 == zeck x `mod` x) [1..100]
main :: IO ()main = print output

Select[Range[4000], Divisible[FromDigits[Flatten[IntegerDigits[#, 2] /. {1 -> {0, 1}}], 2], #] &] (* Amiram Eldar, Jul 08 2019 after Robert G. Wilson v at A048678 *)

A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward.

Original entry on oeis.org

1, 0, 3, 1, 7, 3, 15, 7, 31, 15, 63, 31, 127, 63, 255, 127, 511, 255, 1023, 511, 2047, 1023, 4095, 2047, 8191, 4095, 16383, 8191, 32767, 16383, 65535, 32767, 131071, 65535, 262143, 131071, 524287, 262143, 1048575, 524287, 2097151, 1048575, 4194303, 2097151, 8388607, 4194303

Offset: 0

Jim Singh, Jul 12 2018

			Let 1. The first four terms are 1, (1-1)/2 = 0, 2*1+1 = 3, 1.
Let 4*1+3 = 7. The next four terms are 7, (7-1)/2 = 3, 2*7+1 = 15, 7.
Let 4*7+3 = 31. The next four terms are 31, (31-1)/2 = 15, 2*31+1 = 63, 31; etc.

Jim Singh and others, Fascinating periodic sequence pairs, Mersenne Forum thread, July 2018.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A000225, A024036, A083420.

a:=[1,0,3];; for n in [4..45] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # Muniru A Asiru, Jul 14 2018

seq(coeff(series((1-x+x^2)/((1-x)*(1-2*x^2)), x,n+1),x,n),n=0..45); # Muniru A Asiru, Jul 14 2018

CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - 2 x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Jul 13 2018 *)
LinearRecurrence[{1, 2, -2}, {1, 0, 3}, 46] (* Robert G. Wilson v, Jul 21 2018 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n>2, a(0)=1, a(1)=0, a(2)=3.
From Bruno Berselli, Jul 12 2018: (Start)
G.f.: (1 - x + x^2)/((1 - x)*(1 - 2*x^2)).
a(n) = 2*a(n-2) + 1 for n>1, a(0)=1, a(1)=0.
a(n) = (1 + (-1)^n)*(2^(n/2) - 2^((n-3)/2)) + 2^((n-1)/2) - 1.
Therefore: a(4*k) = 2*4^k - 1, a(4*k+1) = 4^k - 1, a(4*k+2) = 4^(k+1) - 1, a(4*k+3) = 2*4^k - 1. (End)

cp's OEIS Frontend

A303449 Denominator of (2*n+1)/(2^(2*n+1)-1).

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A331895 Positive numbers k such that the binary and negabinary representations of k and the negabinary representation of -k are all palindromic.

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A368275 Fibonacci zig-zag function.

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A232710 Binary numbers (written in decimal) such that the sum of digits mod 2 equals the product of digits mod 2.

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A233868 Numbers that are not the sum of two evil numbers.

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A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.

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A308267 Numbers k such that k divides A048678(k).

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A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward.

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A303449 Denominator of (2n+1)/(2^(2n+1)-1).