A004760 -id:A004760 - cp's OEIS frontend

A151976 Minimal recursive sequence beginning with 5 similar to N with respect to property of integer to be or not to be in A079523.

5, 6, 8, 10, 13, 14, 17, 18, 21, 22, 24, 26, 29, 30, 32, 34, 37, 38, 40, 42, 45, 46, 49, 50, 53, 54, 56, 58, 61

Offset: 1

Vladimir Shevelev, Jul 12 2009

nonn

			a(2)=6 since 6>5 is the minimal integer such that 2 and 6 simultaneously are not in A079523.

V. Shevelev, Several results on sequences which are similar to the positive integers

Cf. A079523 A035263 A010060 A159559 A159560 A004760.

For n>=1, a(n+1)=min{m>a(n): A035263(m)=A035263(n+1)}

A294991 Let S be the sequence of integer sets defined by the following rules: S(0) = {0}, S(1) = {1} and for any k > 0, S(2k) = {2k} U S(k) and S(2k+1) = {2k+1} U S(k) U S(k+1) (where X U Y denotes the union of the sets X and Y); a(n) = the number of elements of S(n).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 12 2017

nonn

For any n >= 0, a(n) corresponds the number of calls to the "fusc" function (defined by Dijkstra) required to compute A002487(n) with an implementation using memoization, and starting with an empty cache.
The sequence A215673 corresponds to the variant without memoization.
For any n > 0, a(n) <= A215673(n) (with equality iff n is a power of 2).
The scatterplot of the ordinal transform of the sequence shows a network of broken lines.
Also: for n >= 1, a(n)+2 is the number of states in the minimal complete deterministic finite automaton that accepts the base-2 representation of m and m+n in parallel, starting with the most significant digit. - Jeffrey Shallit, Jul 22 2023

			The first terms, alongside the corresponding set S(n), are:
n   a(n)    S(n)
--  ----    -----
0   1       { 0 }
1   1       { 1 }
2   2       { 1, 2 }
3   3       { 1, 2, 3 }
4   3       { 1, 2, 4 }
5   4       { 1, 2, 3, 5 }
6   4       { 1, 2, 3, 6 }
7   5       { 1, 2, 3, 4, 7 }
8   4       { 1, 2, 4, 8 }
9   6       { 1, 2, 3, 4, 5, 9 }
10  5       { 1, 2, 3, 5, 10 }
11  6       { 1, 2, 3, 5, 6, 11 }
12  5       { 1, 2, 3, 6, 12 }
13  7       { 1, 2, 3, 4, 6, 7, 13 }
14  6       { 1, 2, 3, 4, 7, 14 }
15  7       { 1, 2, 3, 4, 7, 8, 15 }
16  5       { 1, 2, 4, 8, 16 }
17  8       { 1, 2, 3, 4, 5, 8, 9, 17 }
18  7       { 1, 2, 3, 4, 5, 9, 18 }
19  8       { 1, 2, 3, 4, 5, 9, 10, 19 }
20  6       { 1, 2, 3, 5, 10, 20 }
See also illustration of the first terms in Links section.

Rémy Sigrist, Scatterplot of the first 100000 terms of the ordinal transform of the sequence
Rémy Sigrist, Illustration of the first terms
Index entries for sequences related to Stern's sequences

Cf. A002487, A004760, A070939, A209229, A215673.

a(n) = my (S = Set(n), u = 1); while (u <= #S, my (v = S[#S-u+1]); if (v>1, if (v%2==0, S = setunion(S, Set(v/2)), S = setunion(S, Set([(v-1)/2, (v+1)/2])))); u++;); return (#S)

a(n) = 2*floor(log_2 n) - nu_2(n) + [n is a power of 2] + [1st two bits of n in base 2 are 11] = 2*A000523(n) - A007814(n) + A209229(n) + [n belongs to A004755], for n >= 1. - Jeffrey Shallit, Jul 20 2023
a(2*n) = a(n) + 1, n >= 1.
a(4*n+1) = a(2*n+1)+2, n >= 2.
a(4*n+3) = a(2*n+1)+2, n >= 0.
a(2^k) = k + 1 for any k >= 0.
Empirically: a(2*k-1) = 2*A070939(k) - 2*A209229(k) + [(k-1) is in A004760] for any k > 0 (where [P]=1 if P is true and [P]=0 otherwise).

A295897 Numbers in whose binary expansion there are no 1-runs of odd length followed by a 0 to their right.

Original entry on oeis.org

0, 1, 3, 6, 7, 12, 13, 15, 24, 25, 27, 30, 31, 48, 49, 51, 54, 55, 60, 61, 63, 96, 97, 99, 102, 103, 108, 109, 111, 120, 121, 123, 126, 127, 192, 193, 195, 198, 199, 204, 205, 207, 216, 217, 219, 222, 223, 240, 241, 243, 246, 247, 252, 253, 255, 384, 385, 387, 390, 391, 396, 397, 399, 408, 409, 411, 414, 415, 432

Offset: 1

Antti Karttunen, Dec 01 2017

No runs of 1-bits of odd length allowed in the binary expansion of n (A007088), except that when n is an odd number, then the rightmost run may have an odd length. Subsequence A277335 does not allow that exception.
A005940(1+a(n)) yields a permutation of A028982, squares and twice squares.
Running maximum without repetition of the decimal equivalent of Gray code for n (A003188). - Frédéric Nouvier, Aug 14 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A005940, A028982, A280873.
Subsequence of A004760.
Cf. A277335 (a subsequence).
Cf. A295896 (characteristic function).

[x ^ (x>>1) for x in range(0,2048) if (x & (x<<1) == 0)]
# Frédéric Nouvier, Aug 14 2020

def A295897(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s <<= 1
        if d <= n:
            s += 1
            n -= d
    return s>>1^s # Chai Wah Wu, Apr 25 2025

fn main() {
    for i in (0..2048)
        // Filter to get A003714
        .filter(|n| n & (n << 1) == 0)
        // Map to produce A295897
        .map(|n| n ^ (n >> 1))
    {
        println!("{}", i);
    }
} // Frédéric Nouvier, Aug 14 2020

a(n) = A003714(n-1) XOR ( A003714(n-1) >> 1 ). - Frédéric Nouvier, Aug 14 2020

A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).

Original entry on oeis.org

1, 2, 10, 62, 469, 4028, 37984, 385202, 4144798, 46882400, 553733875, 6795347708, 86314711993, 1131422763410, 15268625617174, 211726229534738, 3012057754693912, 43903115899714844, 654923002676505376, 9989373316478767304, 155663132037403882606, 2476418549848925209424, 40195761790035415573939

Offset: 2

Paul D. Hanna, Feb 13 2025

nonn

Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.

			G.f.: A(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + 46882400*x^11 + 553733875*x^12 + ...
where A(x - A(x)) = x^2/(1 - x^2).
Let B(x) = Series_Reversion(x - A(x)), where
B(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ... + A380678(n)*x^n + ...
then B(x) = x + A(B(x)).

Paul D. Hanna, Table of n, a(n) for n = 2..520

Cf. A380678, A276370, A004760.

/* Generates N terms of this sequence */
N = 40; A=x^2; for(m=1,N, A=truncate(A); B = serreverse(x - A +x*O(x^m)); A = B^2/(1-B^2) ); Vec(A)

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = x^2/(1 - x^2).
(2) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
(3) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n!.

A051464 Number of divisors of 4*(2^n-1) + 1.

Original entry on oeis.org

2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 4, 6, 4, 4, 16, 2, 4, 2, 4, 2, 4, 8, 12, 8, 2, 4, 8, 4, 4, 4, 4, 4, 4, 8, 8, 4, 8, 16, 8, 4, 8, 8, 6, 16, 8, 8, 8, 16, 8, 4, 32, 32, 8, 4, 8, 4, 4, 8, 16, 8, 8, 16, 48, 16, 16, 8, 4, 16, 4, 16, 16, 8, 8, 8, 16, 16, 8, 16, 32

Offset: 1

Edwin D. Evans, eevans2(AT)pacbell.net

nonn

Create a table with tau(2^n-1) as the first row (A046801) and tau(m) as the first column (A000005). The second column is tau(A004760) and so on. Rows 2, 3 and 4 are easily described in terms of row 1. This sequence is row 5.

Sean A. Irvine, Table of n, a(n) for n = 1..591

Cf. A000005, A036563.

Array[DivisorSigma[0, 4*(2^# - 1) + 1] &, 81] (* Michael De Vlieger, Sep 15 2021 *)

a(n) = numdiv(4*(2^n-1) + 1); \\ Michel Marcus, Sep 16 2021

a(n) = tau(4*(2^n -1)+1), where d(n) = A000005(n).

a(81) corrected by Sean A. Irvine, Sep 15 2021

A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).

Original entry on oeis.org

1, 2, 1, 2, 6, 5, 1, 2, 10, 6, 14, 9, 5, 13, 1, 2, 18, 10, 22, 12, 6, 14, 30, 17, 9, 5, 11, 25, 13, 29, 1, 2, 34, 18, 38, 20, 10, 22, 46, 24, 12, 6, 54, 28, 14, 30, 62, 33, 17, 9, 19, 41, 5, 11, 23, 49, 25, 13, 27, 57, 29, 61, 1, 2, 66, 34, 70, 36, 18, 38, 78

Offset: 1

Rémy Sigrist, Nov 01 2017

More informally, the parity of the iterate of the sequence at n gives the binary expansion of n (beyond the leading 1).
Apparently, iterating the sequence always leads to one of these three loops:
- the fixed point (1) iff we start from 2^k-1 for some k > 0,
- the fixed point (2) iff we start from 2^k for some k > 0,
- or (5, 6) for any other starting value.
a(n) is even iff n belongs to A004754.
a(n) is odd iff n belongs to A004760.
If a(n) > n then a(n) = A080541(n).
If n < 2^k then a(n) < 2^k.
Apparently, if a(n) > 2, then A054429(a(n)) = a(A054429(n)); this accounts for the symmetry of the part connected to the loop (5,6) in the oriented graph of this sequence.

			For n=11:
- the binary representation of 11 is (1,0,1,1),
- a(11) = 14 has parity 0,
- a(14) = 13 has parity 1,
- a(13) = 5 has parity 1,
- we find the binary digits of 11 beyond the initial 1, in order: 0, 1, 1.
See also representations of first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Representation of the first 32 terms as an oriented graph (n -> a(n))
Rémy Sigrist, Representation of the first 64 terms as an oriented graph (n -> a(n))
Rémy Sigrist, PARI program for A294523

Cf. A000079, A000225, A000975, A004754, A004760, A052997, A054429, A080541, A081253, A081254, A266248, A266613, A266721, A267045.

PARI
```
See Links section.
```

a(n) = 1 iff n = A000225(k) for some k > 0.
a(n) = 2 iff n = A000079(k) for some k > 0.
a(n) = 5 iff n = A081254(k) for some k > 2.
a(n) = 6 iff n = A000975(k) for some k > 2.
a(n) = 10 iff n = A081253(k) for some k > 2.
a(n) = 12 iff n = A266613(k) for some k > 3.
a(n) = 13 iff n = A052997(k) for some k > 2.
a(n) = 14 iff n = A266721(k) for some k > 2.
a(n) = 18 iff n = A267045(k) for some k > 3.
a(n) = 54 iff n = A266248(k) for some k > 4.
These formulas come from the fact that each sequence on the right side, say f, eventually satisfies: f(n) = floor(f(n+1)/2), and f(n) and f(n+2) have the same parity.

A375378 Value of the power tower formed by the numbers in the n-th composition (in standard order).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 4, 2, 1, 1, 1, 1, 5, 4, 9, 3, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 16, 4, 27, 9, 3, 3, 16, 8, 16, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 25, 5, 64, 16, 4, 4, 81, 27, 81, 9, 3, 3, 3, 3, 32, 16

Offset: 0

Pontus von Brömssen, Aug 14 2024

nonn

It is natural to define a(0) = 1.

			For n = 38, the 38th composition is (3,1,2), so a(38) = 3^1^2 = 3^1 = 3.

Cf. A004760, A053645, A065120, A066099 (compositions in standard order), A375379.

a(n) = A065120(n)^a(A053645(n)) for n >= 1.

a(n) = 1 if and only if A065120(n) <= 1, i.e., if and only if n is in A004760.

A383593 In the binary expansion of n, change the most significant 0 bit to 1, if there is any 0 bit.

Original entry on oeis.org

1, 1, 3, 3, 6, 7, 7, 7, 12, 13, 14, 15, 14, 15, 15, 15, 24, 25, 26, 27, 28, 29, 30, 31, 28, 29, 30, 31, 30, 31, 31, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 56, 57, 58, 59, 60, 61, 62, 63, 60, 61, 62, 63, 62, 63, 63, 63, 96, 97, 98, 99, 100

Offset: 0

Frederik P.J. Vandecasteele, Jun 11 2025

n = 0 is taken to be a single 0 bit, but for all other n no leading 0 bits are used.

The plot of the sequence is fractal.

			a(25) = 29 since 25 = 11001_2 becomes 11101_2 = 29.

Michael S. Branicky, Table of n, a(n) for n = 0..16383

Cf. A000225 (fixed points), A004760 (range of values), A063250.

def a(n): return int(bin(n)[2:].replace('0', '1', 1), 2)
print([a(n) for n in range(70)]) # Michael S. Branicky, Jun 11 2025

def A383593(n): return (n if (t:=bin(n)[2:].find('0'))==-1 else n+(1<Chai Wah Wu, Jun 17 2025

a(n) = n + floor(2^(A063250(n)-1)) for n > 0. - David Radcliffe, Jun 12 2025

A151994 For k=A079523(n),n>=2, let {S_k} be the minimal recursive sequence beginning with k similar to N with respect to property of integer to be or not to be in A079523. Then a(n) is the point of confluence of {S_k} with {S_5}.

Original entry on oeis.org

5, 13, 13, 29, 29, 61, 61, 61, 61

Offset: 2

Vladimir Shevelev, Jul 12 2009

nonn

			Note that, {S_5} is {5,6,8,10,13,...}(see A162736) and {S_7} is {7,8,10,11,13,...}, then a(3)=13.

Cf. A162736 A079523 A159615 A159619 A159559 A159560 A004760 A035263.

cp's OEIS Frontend

A151976 Minimal recursive sequence beginning with 5 similar to N with respect to property of integer to be or not to be in A079523.

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A294991 Let S be the sequence of integer sets defined by the following rules: S(0) = {0}, S(1) = {1} and for any k > 0, S(2*k) = {2*k} U S(k) and S(2*k+1) = {2*k+1} U S(k) U S(k+1) (where X U Y denotes the union of the sets X and Y); a(n) = the number of elements of S(n).

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A295897 Numbers in whose binary expansion there are no 1-runs of odd length followed by a 0 to their right.

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A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).

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A051464 Number of divisors of 4*(2^n-1) + 1.

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Mathematica

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A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).

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A375378 Value of the power tower formed by the numbers in the n-th composition (in standard order).

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A383593 In the binary expansion of n, change the most significant 0 bit to 1, if there is any 0 bit.

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A294991 Let S be the sequence of integer sets defined by the following rules: S(0) = {0}, S(1) = {1} and for any k > 0, S(2k) = {2k} U S(k) and S(2k+1) = {2k+1} U S(k) U S(k+1) (where X U Y denotes the union of the sets X and Y); a(n) = the number of elements of S(n).