A005352 -id:A005352 - cp's OEIS frontend

A320642 Number of 1's in the base-(-2) expansion of -n.

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

Offset: 1

Jianing Song, Oct 18 2018

Number of 1's in A212529(n).
Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(-n), n >= 1. See A027615 for the other half of f.
For k > 1, the earliest occurrence of k is n = A086893(k-1).

			A212529(11) = 110101 which has four 1's, so a(11) = 4.
A212529(25) = 111011 which has five 1's, so a(25) = 5.
A212529(51) = 11011101 which has six 1's, so a(51) = 6.

Jianing Song, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Negabinary.
Eric Weisstein's World of Mathematics, Negadecimal.
Wikipedia, Negative base.

Cf. A000120, A005352, A027615, A086893, A212529.

b[n_] := b[n] = b[Quotient[n - 1, -2]] + Mod[n, 2]; b[0] = 0; a[n_] := b[-n]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

b(n) = if(n==0, 0, b(n\(-2))+n%2)
a(n) = b(-n)

a(n) == -n (mod 3).

a(n) = A000120(A005352(n)). - Michel Marcus, Oct 23 2018

A331821 Positive numbers k such that -k and -(k + 1) are both negabinary-Niven numbers (A331728).

Original entry on oeis.org

2, 3, 8, 9, 15, 24, 27, 32, 33, 39, 54, 55, 63, 77, 111, 114, 115, 123, 128, 129, 135, 144, 159, 174, 175, 203, 234, 235, 245, 255, 264, 294, 295, 329, 370, 371, 384, 413, 414, 415, 444, 447, 474, 475, 495, 504, 507, 512, 513, 519, 534, 535, 543, 580, 581, 624

Offset: 1

Amiram Eldar, Jan 27 2020

			8 is a term since both -8 and -(8 + 1) = -9 are negabinary-Niven numbers: A039724(-8) = 1000 and 1 + 0 + 0 + 0 = 1 is a divisor of 8, and A039724(-9) = 1011 and 1 + 0 + 1 + 1 = 3 is a divisor of 9.

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

Cf. A005352, A027615, A039724, A328209, A328213, A330927, A330931, A331086, A331089, A331728, A331819, A331820.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[negaBinNivenQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s

A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 57, 65, 85, 127, 155, 217, 257, 273, 325, 341, 455, 511, 635, 857, 889, 993, 1025, 1105, 1253, 1285, 1365, 1799, 2047, 2159, 2555, 2667, 3417, 3577, 3641, 3937, 4097, 4161, 4369, 4433, 4965, 5125, 5189, 5397, 5461, 6951, 7175, 7967, 8191

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.

			5 is a term since the negabinary representation of 5, 101, and the negabinary representation of -5, 1111, are both palindromic.

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

Intersection of A331891 and A331892.

Cf. A005352, A039724, A212529.

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[2^13], nbPalinQ]

A331823 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negabinary-Niven numbers (A331728).

Original entry on oeis.org

2, 8, 32, 54, 114, 128, 174, 234, 294, 370, 413, 414, 474, 512, 534, 580, 654, 774, 894, 954, 1000, 1014, 1134, 1430, 1734, 1794, 1840, 1854, 1914, 1974, 2034, 2048, 2093, 2094, 2154, 2214, 2334, 2574, 2680, 2694, 2814, 2870, 3054, 3100, 3520, 3773, 3774, 3834

Offset: 1

Amiram Eldar, Jan 27 2020

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

Cf. A005352, A027615, A039724, A154701, A328210, A328214, A330932, A331087, A331090, A331728, A331819, A331822.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; nConsec = 3; neg = negaBinNivenQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec+1; While[c < 50, If[And @@ neg, c++; AppendTo[seq, k - nConsec]]; neg = Join[Rest[neg], {negaBinNivenQ[k]}]; k++]; seq

cp's OEIS Frontend

A320642 Number of 1's in the base-(-2) expansion of -n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A331821 Positive numbers k such that -k and -(k + 1) are both negabinary-Niven numbers (A331728).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A331823 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negabinary-Niven numbers (A331728).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica