A090252 - OEIS

%I #133 Aug 11 2024 23:41:01

%S 1,2,3,5,4,7,9,11,13,17,8,19,23,25,21,29,31,37,41,43,47,53,16,59,61,

%T 67,71,73,55,79,27,49,83,89,97,101,103,107,109,113,127,131,137,139,

%U 149,151,26,157,163,167,173,179,181,191,193,197,199,211,85,121,223,227,57,229

%N The Two-Up sequence: a(n) is the least positive number not already used that is coprime to the previous floor(n/2) terms.

%C a(n) is coprime to the next n terms. - _David Wasserman_, Oct 24 2005

%C 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.

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

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

%C 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

%C Comments from _N. J. A. Sloane_, May 23 2022: (Start)

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

%C 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, ...

%C 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.

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

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

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

%C For example, from _Michael S. Branicky_'s 10000-term b-file, it appears that:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%C (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.)

%C 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.

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

%C 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.

%C 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.

%C (End)

%C From _N. J. A. Sloane_, Jun 06 2022 (Start)

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

%C   (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.

%C 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.

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

%C 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).

%C (End)

%C 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

%H Michael S. Branicky, <a href="/A090252/b090252.txt">Table of n, a(n) for n = 1..34886</a>

%H Russ Cox, <a href="/A354164/a354164.txt">Table of nonprime entries in A090252: n, A090252(n), # of prime factors, n = 1..3527.</a>

%H Russ Cox, <a href="/A090252/b090252.txt.gz">Table of n, a(n) for n = 1..5764982</a>, up to the first term that is greater than 10^8 [gzipped file]

%H Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, <a href="http://arxiv.org/abs/2209.04108">The Binary Two-Up Sequence</a>, arXiv:2209.04108 [math.CO], Sep 11 2022.

%H N. J. A. Sloane, <a href="https://njas.blog/2022/06/03/the-two-up-sequence-a090252/">Blog post about the Two-Up sequence</a>, June 13 2022.

%H Hugo van der Sanden, <a href="/A249064/a249064.txt">Perl program</a> to calculate this sequence and A249064 (requires Math::Pari)

%H Hugo van der Sanden, <a href="https://github.com/hvds/seq/blob/master/A249064/A090252">Faster Perl program</a> on github, used to compute 10^9 terms. [Link changed by _N. J. A. Sloane_, Jun 19 2022]

%H Hugo van der Sanden, <a href="/A354164/a354164_b.txt">Table of nonprime entries in the first 10^9 terms of A090252</a> [See beginning of the file for description. The blog in the above link has comments from _Hugo van der Sanden_ describing the algorithm used to generate this table.]

%t nn = 120; c[_] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[Array[a[i - #] &, Floor[i/2]], CoprimeQ[#, k] &]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn]] (* _Michael De Vlieger_, May 21 2022 *)

%o (Python)

%o from math import gcd, prod

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     alst = [1]; aset = {1}; yield 1

%o     mink = 2

%o     for n in count(2):

%o         k, prodall = mink, prod(alst[n-n//2-1:n-1])

%o         while k in aset or gcd(prodall, k) != 1: k += 1

%o         alst.append(k); aset.add(k); yield k

%o         while mink in aset: mink += 1

%o print(list(islice(agen(), 64))) # _Michael S. Branicky_, May 21 2022

%o (PARI) A090252_first(N, U=[0], L=List())=vector(N, i, for(k=U[1]+1, oo, setsearch(U, k) && next; foreach(L,m, gcd(k,m)>1 && next(2)); bitand(i,1) || listpop(L,1); listput(L,k); if( k>U[1]+1, U=setunion(U,[k]), U[1]++; while(#U>1 && U[2]==U[1]+1, U=U[^1]));break); L[#L]) \\ _M. F. Hasler_, Jun 14 2022

%Y Cf. A083329, A084937, A196305, A249064, A354146, A354148-A354151, A354154, A354159-A354167, A354255 (even terms), A355893.

%Y See also A354169, A354764, A354765, A355057.

%Y See A247665 for the case when the numbers are required to be at least 2. A353730 is another version.

%Y For a squarefree analog, see A354790, A354791, A354792.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Nov 27 2003

%E More terms from _David Wasserman_, Oct 24 2005