A014550 -id:A014550 - cp's OEIS frontend

A344342 Numbers k such that k and k + 1 are both Gray-code Niven numbers (A344341).

1, 2, 3, 6, 7, 8, 14, 15, 27, 30, 31, 32, 39, 44, 51, 56, 62, 63, 75, 99, 104, 111, 123, 126, 127, 128, 135, 144, 155, 159, 174, 175, 184, 185, 195, 204, 207, 215, 224, 231, 234, 235, 243, 244, 248, 254, 255, 264, 275, 284, 294, 300, 304, 305, 315, 335, 354, 375

Offset: 1

Amiram Eldar, May 15 2021

			1 is a term since 1 and 2 are both Gray-code Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005811, A014550.
Subsequence of: A344341.
Subsequences: A344343 and A344344.
Similar sequences: A330927 (decimal), A328205 (factorial), A328209 (Zeckendorf), A328213 (lazy Fibonacci), A330931 (binary), A331086 (negaFibonacci), A333427 (primorial), A334309 (base phi), A331820 (negabinary), A342427 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[400], And @@ gcNivenQ[# + {0, 1}] &]

A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 2, 6, 7, 14, 30, 31, 62, 126, 127, 174, 184, 234, 243, 254, 304, 474, 483, 510, 511, 534, 543, 544, 783, 784, 903, 904, 954, 963, 1022, 1134, 1144, 1253, 1264, 1448, 1475, 1504, 1895, 1914, 1923, 1974, 2046, 2047, 2093, 2094, 2104, 2814, 2888, 2944, 3054, 3064

Offset: 1

Amiram Eldar, May 15 2021

			1 is a term since 1, 2 and 3 are all Gray-code Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005811, A014550.
Subsequence of A344341 and A344342.
Subsequences: A344344.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary), A342428 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[3000], AllTrue[# + {0, 1, 2}, gcNivenQ] &]

A344344 Starts of runs of 4 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 6, 30, 126, 510, 543, 783, 903, 2046, 2093, 3773, 3903, 7133, 7743, 8190, 8223, 8703, 10087, 12303, 12543, 14343, 14463, 15423, 15903, 16143, 16263, 20167, 22687, 27727, 30247, 30653, 30783, 32766, 35629, 40327, 47509, 47887, 49133, 50407, 57533, 60071, 60487

Offset: 1

Amiram Eldar, May 15 2021

Are there 5 consecutive Gray-code Niven numbers? There are no such numbers below 10^10.

			1 is a term since 1, 2, 3 and 4 are all Gray-code Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005811, A014550.
Subsequence of A344341, A344342 and A344343.
Similar sequences: A141769 (decimal), A328207 (factorial), A328211 (Zeckendorf), A328215 (lazy Fibonacci), A330933 (binary), A334311 (base phi), A331824 (negabinary), A342429 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[60000], AllTrue[# + {0, 1, 2, 3}, gcNivenQ] &]

A050605 Column/row 2 of A050602: a(n) = add3c(n,2).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 3, 3, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 4, 4, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 3, 3, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 5, 5, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 3, 3, 0, 0, 1, 1, 0, 0, 2, 2

Offset: 0

Antti Karttunen, Jun 22 1999

nonn

It seems that (n - Sum_{k=1..n} a(k) )/log(n) is bounded. - Benoit Cloitre, Oct 03 2002
2^a(n-1) is the highest power of 2 dividing the triangular number A000217(n) = n*(n+1)/2, for n >= 1. - Benoit Cloitre, Oct 03 2002 [corrected and rewritten by Wolfdieter Lang, Nov 21 2019]
a(n) is the number of trailing 0's in the binary reflected Gray code of n+1 (A014550). - Amiram Eldar, May 15 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.

Bisection gives column/row 1 of A050602: A007814.

Cf. A000217, A001511, A161737, A069834, A014550.

[Valuation(n*(n+1)/2, 2): n in [1..120]]; // Vincenzo Librandi, Aug 11 2017

with(Bits): add3c := proc(a, b) option remember; `if`(0 = And(a, b), 0, 1 + add3c(Xor(a, b), 2*And(a, b))) end: A050605 := n -> add3c(n, 2):
seq(A050605(n), n=0..80); # Johannes W. Meijer, Jun 18 2009; updated by Peter Luschny, Jul 12 2019

Table[IntegerExponent[(n + 1)(n + 2)/2, 2], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)

PARI
```
a(n)=valuation(n*(n+1)/2,2)
```

a(4*n+2) = A001511(n). - Johannes W. Meijer, Jun 18 2009

a(n) = A007814(n+1) + A007814(n+2) - 1. - Ridouane Oudra, Oct 08 2019

A047521 Numbers that are congruent to {0, 7} mod 8.

Original entry on oeis.org

0, 7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47, 48, 55, 56, 63, 64, 71, 72, 79, 80, 87, 88, 95, 96, 103, 104, 111, 112, 119, 120, 127, 128, 135, 136, 143, 144, 151, 152, 159, 160, 167, 168, 175, 176, 183, 184, 191, 192, 199, 200, 207, 208, 215, 216, 223, 224, 231, 232

Offset: 1

N. J. A. Sloane

Numbers such that the n-th triangular number is divisible by 4. - Charles R Greathouse IV, Apr 07 2011

Except for 0, numbers whose binary reflected Gray code (A014550) ends with 00. - Amiram Eldar, May 17 2021

David Lovler, Table of n, a(n) for n = 1..10000
Lars Pos, Met kleine stapjes grote sprongen make, Pythagoras 61-4. Solutions of returning to the origin after steps of increasing width 1,2,3,.. in the 4 directions on a square grid (in Dutch).
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A004771.

Cf. A014550, A030308, A274406.

{#,#+7}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,7,8},60] (* Harvey P. Dale, Oct 30 2016 *)

a(n) = 4*n - 5/2 + 3*(-1)^n/2; \\ David Lovler, Jul 25 2022

kmax <- 10 # by choice
a <- c(0,7)
for(k in 3:kmax) a <- c(a, a + 2^k)
a
# Yosu Yurramendi, Jan 18 2022

a(n) = 8*n - a(n-1) - 9 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 3*(-1)^n/2 - 5/2 + 4*n.
G.f.: x^2*(7+x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=7 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = log(2)/2 + sqrt(2)*log(sqrt(2)+1)/8 - (sqrt(2)+1)*Pi/16. - Amiram Eldar, Dec 18 2021
E.g.f.: 1 + ((8*x -5)*exp(x) + 3*exp(-x))/2. David Lovler, Aug 22 2022

More terms from Vincenzo Librandi, Aug 06 2010

A047457 Numbers that are congruent to {3, 4} mod 8.

Original entry on oeis.org

3, 4, 11, 12, 19, 20, 27, 28, 35, 36, 43, 44, 51, 52, 59, 60, 67, 68, 75, 76, 83, 84, 91, 92, 99, 100, 107, 108, 115, 116, 123, 124, 131, 132, 139, 140, 147, 148, 155, 156, 163, 164, 171, 172, 179, 180, 187, 188, 195, 196, 203, 204, 211, 212, 219, 220, 227

Offset: 1

N. J. A. Sloane

Union of A017101 and A017113. - Michel Marcus, Feb 25 2014

Numbers whose binary reflected Gray code (A014550) has a single trailing zero. - Amiram Eldar, May 17 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000120, A014550, A017101, A017113, A047456 (superset), A063787.

A047457:=n->(-5 - 3*(-1)^n + 8*n)/2; seq(A047457(n), n=1..100); # Wesley Ivan Hurt, Mar 04 2014

Table[(-5 - 3*(-1)^n + 8*n)/2, {n, 100}] (* Wesley Ivan Hurt, Mar 04 2014 *)
Flatten[Table[8n + {3, 4}, {n, 0, 29}]] (* Alonso del Arte, Mar 04 2014 *)

a(n) = 8*n - a(n-1) - 9 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3+x+4*x^2)/((1-x)^2*(1+x)). - Colin Barker, May 13 2012
a(n) = (-5 - 3*(-1)^n + 8*n)/2. - Colin Barker, May 14 2012
A000120(a(n)-1) = A000120(a(n)+1) = A063787(n). - Ilya Lopatin and Juri-Stepan Gerasimov, Feb 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A280998 Numbers with a prime number of 1's in their binary reflected Gray code representation.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103

Offset: 1

Indranil Ghosh, Jan 12 2017

From Emeric Deutsch, Jan 28 2018: (Start)
Also the indices of the compositions that have a prime number of parts. For the definition of the index of a composition see A298644.
For example, 27 is in the sequence since its binary form is 11011 and the composition [2,1,2] has 3 parts.
On the other hand, 58 is not in the sequence since its binary form is 111010 and the composition [3,1,1,1] has 4 parts.
The command c(n) from the Maple program yields the composition having index n. (End)

			27 is in the sequence because the binary reflected Gray code representation of 27 is 10110 which has 3 1's, and 3 is prime.

Indranil Ghosh, Table of n, a(n) for n = 1..10001
Wikipedia, Gray code.

Cf. A014550, A052294, A005811, A298644, A101211.

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]:
for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1:
r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc:
RunLengths := proc (L) map(nops, Runs(L)) end proc:
c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc:
A := {}: for n to 175 do if isprime(nops(c(n))) = true then A := `union`(A, {n}) else end if end do: A;
# most of the program is due to W. Edwin Clark. # Emeric Deutsch, Jan 28 2018

Select[Range[100], PrimeQ[DigitCount[BitXor[#, Floor[#/2]], 2, 1]] &] (* Amiram Eldar, May 01 2021 *)

is(n)=isprime(hammingweight(bitxor(n, n>>1))) \\ Charles R Greathouse IV, Jan 12 2017

A280999 Numbers with a prime number of 0's in their binary reflected Gray code representation.

Original entry on oeis.org

7, 8, 12, 14, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 30, 33, 34, 35, 36, 38, 39, 40, 44, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 61, 63, 64, 66, 68, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81, 83, 87, 88, 89, 91, 92, 93, 94, 96, 98, 100, 101, 102, 104, 105, 107

Offset: 1

Indranil Ghosh, Jan 12 2017

			27 is in the sequence because the binary reflected Gray Code representation of 27 is 10110 which has 2 0's and 2 is prime.

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

Cf. A014550, A144754, A280998.

Select[Range[100], PrimeQ[DigitCount[BitXor[#, Floor[#/2]], 2, 0]] &] (* Amiram Eldar, May 01 2021 *)

is(n)=isprime(logint(n,2)-hammingweight(bitxor(n, n>>1))+1) \\ Charles R Greathouse IV, Jan 13 2017

A281379 Numbers which are palindromic in their binary reflected Gray code representation.

Original entry on oeis.org

0, 1, 2, 5, 6, 10, 14, 18, 21, 25, 30, 34, 42, 54, 62, 66, 77, 85, 90, 102, 105, 113, 126, 130, 146, 170, 186, 198, 214, 238, 254, 258, 285, 301, 306, 330, 341, 357, 378, 390, 409, 425, 438, 462, 465, 481, 510, 514, 546, 594, 626, 650, 682, 730, 762, 774, 806, 854, 886, 910, 942

Offset: 1

Indranil Ghosh, Jan 21 2017

A281378 is a subsequence of this sequence.

			34 is in the sequence because the binary reflected Gray code representation of 34 is '110011', which is palindromic.

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

Cf. A006995, A014550, A281378.

Select[Range[0, 10^3], Reverse@ # == # &@ Abs[Prepend[Most@ #, 0] - #] &@
IntegerDigits[#, 2] &] (* Michael De Vlieger, Jan 21 2017 *)

def G(n):
    return bin(n^(n//2))[2:]
i=0
j=1
while j<=10000:
    if G(i)==G(i)[::-1]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1

A281388 Write n in binary reflected Gray 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, 0, 1, 2, 0, 1, 1, 2, 2, 0, 3, 4, 1, 1, 1, 2, 2, 2, 6, 4, 0, 3, 3, 4, 4, 1, 5, 5, 1, 1, 1, 2, 2, 2, 7, 7, 2, 6, 6, 4, 4, 0, 5, 8, 3, 3, 3, 4, 4, 4, 9, 6, 1, 5, 5, 5, 5, 1, 6, 6, 1, 1, 1, 2, 2, 2, 8, 8, 2, 7, 7, 7, 7, 2, 8, 12, 6, 6, 6, 4, 4, 4, 10, 6, 0, 5, 5, 8, 8, 3, 9, 9, 3, 3, 3

Offset: 1

Indranil Ghosh, Jan 21 2017

			For n = 11, the binary reflected Gray code for 11 is '1110'. In '1110', the position of '1' followed immediately to the right by '0' counting from left is 3. So, a(11) = 3.
For n = 12, the binary reflected Gray code for 12 is '1010'. In '1010', the positions of '1' followed immediately to the right by '0' counting from left are 1 and 3. So, a(12) = 1 + 3 = 4.

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

Cf. A003188, A014550, A049501.

def g(n):
    return bin(n^(n//2))[2:]
def a(n):
    x=g(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(A003188(n)).

cp's OEIS Frontend

A344342 Numbers k such that k and k + 1 are both Gray-code Niven numbers (A344341).

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A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

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A344344 Starts of runs of 4 consecutive Gray-code Niven numbers (A344341).

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A050605 Column/row 2 of A050602: a(n) = add3c(n,2).

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A047521 Numbers that are congruent to {0, 7} mod 8.

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A047457 Numbers that are congruent to {3, 4} mod 8.

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A280998 Numbers with a prime number of 1's in their binary reflected Gray code representation.

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A280999 Numbers with a prime number of 0's in their binary reflected Gray code representation.

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