A070939 -id:A070939 - cp's OEIS frontend

A272756 a(n) is the least k such that k > A070939(n * k).

3, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Peter Kagey, May 05 2016

			a(1) = 3 because 3 > A070939(1 * 3) = 2.
a(2) = 5 because 5 > A070939(2 * 5) = 4.
a(5) = 6 because 6 > A070939(5 * 6) = 5.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A070939.

Table[SelectFirst[Range[2^12], # > IntegerLength[n #, 2] &], {n, 80}] (* Michael De Vlieger, May 05 2016, Version 10 *)

A334796 a(n) = (A070939(A334769(n)) - A334770(n))/3.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 4, 4, 4, 4, 2, 3, 3, 4, 4, 3, 3, 4, 4, 3, 5, 5, 5, 5, 3, 2, 4, 4, 4, 4, 2, 3, 5, 5, 5, 5, 5, 3, 5, 2, 5, 4, 5, 4, 4, 5, 4, 5, 2, 5, 3, 5, 6, 6, 6, 6, 3, 4, 5, 5, 4, 3, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 3, 4, 5

Offset: 1

Michael De Vlieger, May 12 2020

nonn

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(m) applied recursively until we reach a single bit.
A334556 is the sequence of rotationally symmetrical T(m).
A central zero-triangle (CZT) is a field of contiguous 0-bits, listed in A334769, a subset of A334556. CZTs have side length k = A334770(n), surrounded on all sides by a layer of 1 bits, and generally j > 1 bits of any parity.
This sequence describes the "frame width" j.
Smallest n with a given value of j appears in A334836. - Michael De Vlieger, May 20 2020

			a(4) pertains to T(599), with A334770(4) = 4.
(1 + A070939(599) - 4)/3 = (1 + 9 - 4)/3 = 6/3 = 2, thus a(4) = 2.
(Diagram, replacing 0 with “.”):
  1 . . 1 . 1 . 1 1 1
   1 . 1 1 1 1 1 . .
    1 1 . . . . 1 .
     . 1 . . . 1 1
      1 1 . . 1 .
       . 1 . 1 1
        1 1 1 .
         . . 1
          . 1
           1
a(11) pertains to T(2359), with A334770(11) = 3.
(1 + A070939(2359) - 4)/3 = (1 + 11 - 3)/3 = 9/3 = 3, thus a(11) = 3.
(Diagram):
  1 . . 1 . . 1 1 . 1 1 1
   1 . 1 1 . 1 . 1 1 . .
    1 1 . 1 1 1 1 . 1 .
     . 1 1 . . . 1 1 1
      1 . 1 . . 1 . .
       1 1 1 . 1 1 .
        . . 1 1 . 1
         . 1 . 1 1
          1 1 1 .
           . . 1
            . 1
             1
From _Michael De Vlieger_, May 14 2020: (Start)
Linear recurrences that produce XOR-triangles with frame length j (table may not be exhaustive):
j   LR          Lower               Upper
-----------------------------------------------------
2   (5, -4)     {39, 151}           {57, 223}
3   (17, -16)   {543, 8607}         {993, 15969}
                {1379, 22115}       {1589, 25397}
                {1483, 23755}       {1693, 27037}
                {2359, 37687}       {3785, 60617}
4   (17, -16)   {22243, 356067}     {25525, 408501}
                {39047, 624775}     {57625, 921881}
                {40679, 650983}     {59257, 948089}
                {171475, 2743763}   {208613, 3337957}
                {356067, 5697251}   {408501, 6536117}
... (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.
Michael De Vlieger, Diagram montage showing XOR-triangles for terms in certain linear recurrences and their bit-reversals, illustrating relations in their appearance, most significantly, constant frame width.
Index entries for sequences related to binary expansion of n
Index entries for sequences related to XOR-triangles
Michael De Vlieger, Diagram montage showing the first dozen XOR-triangles exhibiting frame widths of 2, 3, 4, ..., 12 by row.

Cf. A038554, A070939, A334556, A334769, A334770, A334836.

Block[{f, s = Rest[Import["https://oeis.org/A334556/b334556.txt", "Data"][[All, -1]] ]}, f[n_] := NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]; Array[Block[{n = s[[#]]}, If[# == 0, Nothing, (1 + Floor@ Log2[n] - #)/3] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, f[n][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]] ],r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Length@ s - 1, 2] ]

A345927 Alternating sum of the binary expansion of n (row n of A030190). Replace 2^k with (-1)^(A070939(n)-k) in the binary expansion of n (compare to the definition of A065359).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, -1, 1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, -1, 1, 0, 2, 1, 3, 2, 1, 0, 2, 1, 0, -1, 1, 0, -1, -2, 0, -1, 1, 0, 2, 1, 0, -1, 1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2

Offset: 0

Gus Wiseman, Jul 14 2021

sign

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The binary expansion of 53 is (1,1,0,1,0,1), so a(53) = 1 - 1 + 0 - 1 + 0 - 1 = -2.

Binary expansions of each nonnegative integer are the rows of A030190.
The positions of 0's are A039004.
The version for prime factors is A071321 (reverse: A071322).
Positions of first appearances are A086893.
The version for standard compositions is A124754 (reverse: A344618).
The version for prime multiplicities is A316523.
The version for prime indices is A316524 (reverse: A344616).
A003714 lists numbers with no successive binary indices.
A070939 gives the length of an integer's binary expansion.
A103919 counts partitions by sum and alternating sum.
A328594 lists numbers whose binary expansion is aperiodic.
A328595 lists numbers whose reversed binary expansion is a necklace.
Cf. A000037, A000070, A000120, A027187, A028260, A065359, A069010, A116406, A121016, A191232, A245563, A344609, A344619.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[ats[IntegerDigits[n,2]],{n,0,100}]

a(n) = subst(Pol(Vecrev(binary(n))), x, -1); \\ Michel Marcus, Jul 19 2021

def a(n): return sum((-1)**k for k, bi in enumerate(bin(n)[2:]) if bi=='1')
print([a(n) for n in range(84)]) # Michael S. Branicky, Jul 19 2021

a(n) = (-1)^(A070939(n)-1)*A065359(n).

A264982 Binary width of terms produced by match-making permutation: a(n) = A070939(A266195(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 6, 4, 5, 5, 5, 6, 4, 6, 5, 7, 5, 5, 6, 5, 7, 5, 7, 5, 6, 6, 7, 6, 6, 6, 7, 5, 8, 5, 8, 5, 8, 5, 8, 5, 9, 6, 7, 6, 7, 6, 8, 6, 7, 7, 8, 7, 7, 7, 8, 7, 8, 8, 8, 6, 8, 7, 8, 7, 9, 6, 9, 6, 9, 6, 9, 6, 10, 6, 9, 6, 9, 6, 10, 6, 10, 6, 9, 7, 8, 7, 9, 7, 10, 6, 10, 6, 10, 6, 11, 6, 11, 6, 11, 6, 11, 6, 11, 6, 11, 6, 12, 7

Offset: 1

Antti Karttunen, Dec 26 2015

Each n occurs A000079(n-1) = 2^(n-1) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..2694

Cf. A000079, A070939, A266195.
Cf. A266186 (where n first appears, most likely also the positions of records).
Cf. A266187 (where n last appears).
Cf. A266197 (gives numbers n where a(n) = a(n+1)).

(define (A264982 n) (A070939 (A266195 n)))

a(n) = A070939(A266195(n)).

A268727 One-based index of the toggled bit between n and A268717(n+1): a(n) = A070939(A003987(n,A268717(1+n))).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 13 2016

A fractal sequence like A268726.

Antti Karttunen, Table of n, a(n) for n = 0..8191

One more than A268726.
Cf. A003188, A006068.
Cf. A001511, A003987, A010059, A070939, A101080, A268717.
Cf. also array A268833.

(define (A268727 n) (A070939 (A003987bi n (A268717 (+ 1 n)))))  ;; A003987bi implements the bitwise-XOR, defined in A003987.

a(n) = A001511(1+A006068(n)).
a(n) = A070939(A003987(n,A268717(1+n))).
a(n) = 1 + floor(log_2(n XOR A003188(1+A006068(n)))).
a(n) = A001511(n)*(1-A010059(n)) + 1. - Alan Michael Gómez Calderón, Jun 15 2025

A372097 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new minimum.

Original entry on oeis.org

0, 2, 4, 7, 16, 24, 40, 49, 53, 102, 104, 126, 174, 226, 379, 768, 831, 832, 1439, 1452, 1914, 2291, 2731, 3000, 3363, 3472, 5608, 5883, 6725, 6787, 7438, 8786, 10280, 11948, 12190, 13135, 15170, 15645, 22407, 26232, 27099, 32773, 33085, 40189, 40523, 48068, 51187

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

These are the k-values of the lower envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number. The corresponding negated differences are given in A372098.

Hugo Pfoertner, Table of n, a(n) for n = 1..99
Hugo Pfoertner, Illustration of scatter band bounded by lower and upper records, up to exponents k=8*10^6.

Cf. A000120, A000244, A011754, A070939, A078839, A372098, A372099, A372100.

a372097(upto) = {my (dm=-oo); for (k=0, upto, my (p=3^k, h=hammingweight(p), b=#binary(p)/2,d=b-h); if (d>dm, print1(k,", "); dm=d))};
a372097(60000)

A372098 a(n) = A070939(3^k) - 2*A000120(3^k) with k = A372097(n).

Original entry on oeis.org

-1, 0, 1, 2, 4, 7, 8, 12, 15, 18, 25, 26, 30, 51, 75, 78, 84, 129, 133, 148, 170, 180, 183, 189, 209, 265, 279, 285, 287, 336, 369, 388, 406, 412, 445, 469, 496, 581, 711, 737, 741, 742, 873, 939, 994, 1044, 1078, 1111, 1157, 1158, 1492, 1636, 1767, 1914, 1933

Offset: 1

Hugo Pfoertner, Apr 25 2024

sign

a(n)/2 are the negated differences at supporting points of the lower envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number.

Hugo Pfoertner, Table of n, a(n) for n = 1..99

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372099, A372100.

A372099 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new maximum.

Original entry on oeis.org

0, 1, 3, 5, 11, 27, 71, 119, 140, 158, 198, 218, 441, 537, 538, 868, 1092, 2128, 2294, 2343, 2811, 2911, 3849, 4003, 4655, 5079, 5279, 5920, 6269, 6603, 10181, 10574, 12801, 12803, 15563, 15784, 16054, 16253, 17127, 18257, 20187, 21934, 34633, 49209, 76791, 78938

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

These are the k-values of the upper envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number. The corresponding differences are given in A372100.

Hugo Pfoertner, Table of n, a(n) for n = 1..96
Hugo Pfoertner, Illustration of scatter band bounded by lower and upper records, up to exponents k=8.5*10^6.

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372098, A372100.

a372099(upto) = {my(dm=oo); for (k=0, upto, my (p=3^k, h=hammingweight(p), b=#binary(p)/2, d=b-h); if (d

A372100 a(n) = 2*A000120(3^k) - A070939(3^k) with k = A372099(n).

Original entry on oeis.org

1, 2, 3, 4, 8, 17, 23, 29, 38, 39, 44, 56, 57, 58, 91, 114, 145, 147, 156, 168, 182, 208, 219, 239, 277, 297, 300, 307, 331, 360, 367, 442, 452, 477, 487, 492, 507, 513, 568, 571, 614, 893, 963, 1275, 1283, 1288, 1440, 1563, 1702, 1957, 2019, 2440, 2471, 2566, 3004

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

a(n)/2 are the differences at supporting points of the upper envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number.

Hugo Pfoertner, Table of n, a(n) for n = 1..96

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372098, A372099.

A232237 Primes p such that p-2 and q are primes, where q is concatenation of binary representations of p and p-2: q = p * 2^L + p-2, where L is the length of binary representation of p-2: L=A070939(p-2).

Original entry on oeis.org

5, 7, 31, 271, 283, 433, 1291, 1321, 1429, 1489, 1951, 4723, 5503, 6091, 6133, 6553, 6871, 16651, 16981, 17029, 17191, 17209, 17749, 17791, 18541, 18919, 19471, 20149, 20479, 20551, 20809, 21319, 21649, 21739, 22111, 25309, 25801, 27061, 27409, 27541, 27691, 28549, 29131

Offset: 1

Alex Ratushnyak, Nov 20 2013

			5 is 101 in binary, 3 is 11, and because 10111 = 23d is a prime, 5 is in the sequence.

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

Cf. A000040, A070939, A232235.

Select[Partition[Prime[Range[3200]],2,1],#[[2]]-#[[1]]==2&&PrimeQ[ FromDigits[ Join[IntegerDigits[#[[2]],2],IntegerDigits[#[[1]],2]],2]]&][[All,2]] (* Harvey P. Dale, Feb 25 2018 *)

A232235(n) = a(n) * 2^A070939(a(n)-2) + a(n)-2.

cp's OEIS Frontend

A272756 a(n) is the least k such that k > A070939(n * k).

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A334796 a(n) = (A070939(A334769(n)) - A334770(n))/3.

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A345927 Alternating sum of the binary expansion of n (row n of A030190). Replace 2^k with (-1)^(A070939(n)-k) in the binary expansion of n (compare to the definition of A065359).

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A264982 Binary width of terms produced by match-making permutation: a(n) = A070939(A266195(n)).

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A268727 One-based index of the toggled bit between n and A268717(n+1): a(n) = A070939(A003987(n,A268717(1+n))).

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A372097 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new minimum.

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A372098 a(n) = A070939(3^k) - 2*A000120(3^k) with k = A372097(n).

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A372099 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new maximum.

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A372100 a(n) = 2*A000120(3^k) - A070939(3^k) with k = A372099(n).

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