A090252 The Two-Up sequence: a(n) is the least positive number not already used that is coprime to the previous floor(n/2) terms.

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, 67, 71, 73, 55, 79, 27, 49, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 26, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 85, 121, 223, 227, 57, 229

Offset: 1

Author

Amarnath Murthy, Nov 27 2003

Keywords

• nonn

Comments

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

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..34886
• Russ Cox, Table of nonprime entries in A090252: n, A090252(n), # of prime factors, n = 1..3527.
• Russ Cox, Table of n, a(n) for n = 1..5764982, up to the first term that is greater than 10^8 [gzipped file]
• Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
• N. J. A. Sloane, Blog post about the Two-Up sequence, June 13 2022.
• Hugo van der Sanden, Perl program to calculate this sequence and A249064 (requires Math::Pari)
• Hugo van der Sanden, Faster Perl program on github, used to compute 10^9 terms. [Link changed by _N. J. A. Sloane_, Jun 19 2022]
• Hugo van der Sanden, Table of nonprime entries in the first 10^9 terms of A090252 [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.]

Crossrefs

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

See also A354169, A354764, A354765, A355057.

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

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

Programs

Mathematica

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 *)

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

Python

from math import gcd, prod
from itertools import count, islice
def agen(): # generator of terms
    alst = [1]; aset = {1}; yield 1
    mink = 2
    for n in count(2):
        k, prodall = mink, prod(alst[n-n//2-1:n-1])
        while k in aset or gcd(prodall, k) != 1: k += 1
        alst.append(k); aset.add(k); yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, May 21 2022

Extensions

More terms from David Wasserman, Oct 24 2005