cp's OEIS Frontend

By definition of A084937 this cannot contain two consecutive 0's. So the 0's divide this sequence into runs of 1's and 2's. In the first 100000 terms the following runs of 1's and 2's appear: 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1112, 1121, 1122, 1212, 1221, 2111, 2112, 2121, 2211, 2212, 2221. There is no obvious structure. - N. J. A. Sloane, Nov 07 2014

gcd(A084937(n), A084937(n+1)) = gcd(A084937(n), A084937(n+2)) = 1 by definition of A084937.

A binary m-sequence: expansion of reciprocal of x^2+x+1 (mod 2).

A Chebyshev transform of the Jacobsthal numbers A001045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Feb 16 2004

This is the r = 1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

This is the Fibonacci sequence (A000045) modulo 2. - Stephen Jordan (sjordan(AT)mit.edu), Sep 10 2007

For n > 0: a(n) = A084937(n-1) mod 2. - Reinhard Zumkeller, Dec 16 2007

This is also the Lucas numbers (A000032) mod 2. In general, this is the parity of any Lucas sequence associated with any pair (P,Q) when P and Q are odd; i.e., a(n) = U_n(P,Q) mod 2 = V_n(P,Q) mod 2. See Ribenboim. - Rick L. Shepherd, Feb 07 2009

Starting with offset 1: (1, 1, 0, 1, 1, 0, ...) = INVERTi transform of the tribonacci sequence A001590 starting (1, 2, 3, 6, 11, 20, 37, ...). - Gary W. Adamson, May 04 2009

From Reinhard Zumkeller, Nov 30 2009: (Start)

Characteristic function of numbers coprime to 3.

a(n) = 1 - A079978(n); a(A001651(n)) = 1; a(A008585(n)) = 0;

A000212(n) = Sum_{k=0..n} a(k)*(n-k). (End)

Sum_{k>0} a(k)/k/2^k = log(7)/3. - Jaume Oliver Lafont, Jun 01 2010

The sequence is the principal Dirichlet character of the reduced residue system mod 3 (the other is A102283). Associated Dirichlet L-functions are L(2,chi) = Sum_{n>=1} a(n)/n^2 = 4*Pi^2/27 = A214549, and L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.157536... = -(psi''(1/3) + psi''(2/3))/54 where psi'' is the tetragamma function. [Jolley eq 309 and arXiv:1008.2547, L(m = 3, r = 1, s)]. - R. J. Mathar, Jul 15 2010

a(n+1), n >= 0, is the sequence of the row sums of the Riordan triangle A158454. - Wolfdieter Lang, Dec 18 2010

Removing the first two elements and keeping the offset at 0, this is a periodic sequence (1, 0, 1, 1, 0, 1, ...). Its INVERTi transform is (1, -1, 2, -2, 2, -2, ...) with period (2,-2). - Gary W. Adamson, Jan 21 2011

Column k = 1 of triangle in A198295. - Philippe Deléham, Jan 31 2012

The set of natural numbers, A000027: (1, 2, 3, ...); is the INVERT transform of the signed periodic sequence (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013

Any integer sequence s(n) = |s(n-1) - s(n-2)| (equivalently, max(s(n-1), s(n-2)) - min(s(n-1), s(n-2))) for n > i + 1 with s(i) = j and s(i+1) = k, where j and k are not both 0, is or eventually becomes a multiple of this sequence, namely, the sequence repeat gcd(j, k), gcd(j, k), 0 (at some offset). In particular, if j and k are coprime, then s(n) is or eventually becomes this sequence (see, e.g., A110044). - Rick L. Shepherd, Jan 21 2014

For n >= 1, a(n) is also the characteristic function for rational g-adic integers (+n/3)A001651).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/3)_g for all integers g >= 2 without a factor 3 (A001651). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014

Characteristic function for A007908(n+1) being divisible by 3. a(n) = bit flipped A007908(n+1) (mod 3) = bit flipped A079978(n). - Wolfdieter Lang, Jun 12 2017

Also Jacobi or Kronecker symbol (n/9) (or (n/9^e) for all e >= 1). - Jianing Song, Jul 09 2018

The binomial trans. is 0, 1, 3, 6, 11, 21, 42, 85, 171, 342,.. (see A024495). - R. J. Mathar, Feb 25 2023

This is conjectured and designed to be a permutation of the nonnegative integers, therefore the offset is taken to be zero.

Restricted to positive indices, this is a sequence of positive integers having the same property, then conjectured to be a permutation of the positive integers. (The word "odd" can be omitted in this case.)

If the word "odd" is dropped from the original definition, the sequence starts (0, 1, 3, 6, 2, 7), and then continues from a(6) = 4 onward as the present sequence. This is again conjectured to be a permutation of the nonnegative integers, and a permutation of the positive integers when restricted to the domain [1..oo). The latter however no longer has the property of lexicographic minimality.

See the OEIS wiki page for further considerations about existence, surjectivity and variants. - M. F. Hasler, Nov 24 2019

a(n) is coprime to the next n terms. - David Wasserman, Oct 24 2005

All values up to a(1000000) are either prime powers or semiprimes; this suggests the sequence is unlikely to be a permutation of the integers.

It appears that a(n) is even iff n = 3*2^k-1 for some k (A083356). - N. J. A. Sloane, Nov 01 2014

The even terms in the present sequence are listed in A354255.

We have a(1) = 1 and a(2) = 2. At step k >= 2, the sequence is extended by adding two terms: a(2*k-1) = smallest unused number which is relatively prime to a(k), a(k+1), ..., a(2*k-2), and a(2*k) = smallest unused number which is relatively prime to a(k), a(k+1), ..., a(2*k-1). So at step k=2 we add a(3)=3, a(4)=5; at step k=3 we add a(5)=4, a(6)=7; and so on. - N. J. A. Sloane, May 21 2022

Comments from N. J. A. Sloane, May 23 2022: (Start)

Conjecture 1. A090252 is a subsequence of A354144 (prime powers and semiprimes).

Conjecture 2. The terms of A354144 that are missing from A090252 are 6, 10, 14, 15, 22, 33, 34, 35, 38, 39, 46, 51, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 209, 213, 214, 215, 218, 219, 221, ...

But since there is no proof that any one of these numbers is really missing, this list cannot yet have an entry in the OEIS.

Let S_p = list of indices of terms in A090252 that are divisible by the prime p.

Conjecture 3. For a prime p, there are constants v_1, v_2, ..., v_K and c such that

S_p = { v_1, v_2, ..., v_k, lambda*2^i - 1, i >= c}.

For example, from Michael S. Branicky's 10000-term b-file, it appears that:

S_2 = { 3*2^k-1, k >= 0 } cf. A083329

S_3 = { 2^k-1, k >= 2 } cf. A000225

S_5 = { 4 then 15*2^k-1 k >= 0 } cf. A196305

S_7 = { 6, 15, then 33*2^k-1, k >= 0 }

S_11 = { 8, 29, then 61*2^k-1, k >= 0 }

S_13 = { 9, 47, 97*2^n-1, n >= 0 }

S_17 = { 10, 59, 121*2^n-1, n >= 0 }

S_19 = { 12, 63, 129*2^n-1, n >= 0 }

S_23 = { 13, 65, 133*2^n-1, n >= 0 }

S_29 = { 16, 121, 245*2^n-1, n >= 0 }

S_31 = { 17, 131, 265*2^n-1, n >= 0 }

The initial primes p and the corresponding values of lambda are:

p: 2 3 5 7 11 13 17 19 23 29 31

lambda:..3...1..15..33...61...97..121..129..133..245..265

(This sequence of lambdas does not seem to have any simpler explanation, is not in the OEIS, and cannot be since the terms shown are all conjectural.)

Conjecture 2 is a consequence of Conjecture 3. For example, 6 does not appear in A090252, since the sets S_2 and S_3 are disjoint.

Also 10 does not appear, since S_2 and S_5 are disjoint.

In fact 2*p for 3 <= p <= 11 does not appear, but 26 = 2*13 does appear since S_2 and S_13 have 47 in common.

Assuming the numbers that appear to be missing (see Conjecture 2) really are missing, the numbers that take a record number of steps to appear are 1, 2, 3, 4, 7, 8, 16, 26, 32, 64, 128, 206, 256, 478, 512, 933, ..., and the indices where they appear are 1, 2, 3, 5, 6, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 8191, .... These two sequences are not yet in the OEIS, and cannot be added since the terms are all conjectural.

(End)

From N. J. A. Sloane, Jun 06 2022 (Start)

Theorem: (a) a(n) <= prime(n-1) for all n >= 2 (cf. A354154).

(b) A stronger upper bound is the following. Let c(n) = A354166(n) denote the number of nonprime terms among a(1) .. a(n). Note c(1)=1. Then a(n) <= prime(n-c(n)) for n <> 7 and 14.

It appears that a(n) = prime(n-c(n)) for almost all n. That is, this is the equation to the line in the graph that contains most of the terms.

For example, a(34886) = 408710 (see the b-file) = prime(34886 - A354166(34886)) = prime(34886 - 374) = prime(34512) = 408710.

Another example: Consider Russ Cox's table of the first N = 5764982 terms. We see that a(5764982) = 99999989 = prime(5761455) = prime(N - 3527) which agrees with c(N) = 3527 (from the first Russ Cox link).

(End)

If we consider the May 23 2022 comment, note the conjectured indices show near complete overlap with terms of A081026: 1, 2, 3, 5, 6, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 8191. - Bill McEachen, Aug 09 2024