A355057 -id:A355057 - cp's OEIS frontend

A354765 a(n) is a binary encoded version of A355057(n).

0, 0, 1, 3, 6, 7, 13, 15, 27, 59, 122, 123, 243, 499, 501, 511, 1007, 2031, 4047, 8143, 16271, 32655, 65422, 65423, 130831, 261903, 523791, 1048079, 2096651, 2096671, 4193813, 4193815, 4193311, 8387615, 16775199, 33552415, 67104799, 134213663, 268427295, 536862751, 1073725471, 2147467295, 4294934559, 8589901855, 17179803679

Offset: 1

Michael De Vlieger and N. J. A. Sloane, Jun 18 2022

Let plist = list of forbidden primes for A090252(n); A355057(n) is the product of these primes. Then a(n) = Sum of 2^(i-1) over all prime(i) in plist.

Conversely, if a(n) has binary expansion a(n) = Sum b(i)*2^i, b(i) = 0 or 1, then plist consists of {prime(i+1) such that b(i) = 1}.

			For n = 7 the forbidden primes are 2, 5, 7 = prime(1), prime(3) and prime(4). Their product is A355057(7) = 70. Then a(7) = 2^0 + 2^2 + 2^3 = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A090252, A355057.

# To get first M terms:
with(numtheory);
M:=20; ans:=[0,0,1];
for i from 4 to M do
S:={}; j1:=floor((i+1)/2); j2:=i-1;
  for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:
plis:=sort(convert(S,list));
t3:=0; for ii from 1 to nops(plis) do p:=plis[ii]; p2:=pi(p); t3:=t3+2^(p2-1); od:
ans:=[op(ans),t3];
od:
ans;

from math import gcd, lcm
from itertools import count, islice
from collections import deque
from sympy import primepi, primefactors
def A354765_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 0
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                yield sum(2**(primepi(p)-1) for p in primefactors(b))
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354765_list = list(islice(A354765_gen(),20)) # Chai Wah Wu, Jun 18 2022

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.

Original entry on oeis.org

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

Amarnath Murthy, Nov 27 2003

nonn

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

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.]

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.

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 *)

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

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

More terms from David Wasserman, Oct 24 2005

A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

Original entry on oeis.org

1, 1, 2, 6, 15, 60, 140, 1260, 2772, 36036, 153153, 1225224, 3325608, 76488984, 212469400, 4461857400, 11763078600, 364655436600, 1037865473400, 42552484409400, 107632754682600, 5058739470082200, 33514148989294575, 536226383828713200, 1665124033994425200

Offset: 1

Rémy Sigrist, Jun 06 2022

nonn

The prime divisors of a(n) are forbidden in A090252(n).

			a(7) = A090252(4) * A090252(5) * A090252(6) = 5 * 4 * 7 = 140.

Michael De Vlieger, Table of n, a(n) for n = 1..587
Rémy Sigrist, Representation of the prime divisors of a(n) (blue pixels) and A090252(n) (red pixels)
Rémy Sigrist, PARI program

A355057 is another version.

Cf. A090252, A354757.

With[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 25, -1]]}, Table[Product[s[[k]], {k, Ceiling[n/2], n - 1}], {n, Length[s]}]] (* Michael De Vlieger, Aug 28 2022 *)

PARI
```
See Links section.
```

gcd(a(n), A090252(n)) = 1.

A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

Original entry on oeis.org

2, 12, 27, 115, 252, 1004, 2013, 4031, 16307, 32631, 65279, 261375, 524270, 2096110, 4194253, 8386527, 16773119, 67096575, 134217659, 536854459, 1073741623, 2147450751, 4294901759, 17179672575, 34359737599, 137438690559, 274877382143, 549754765311, 2199022205950, 4398044412927, 8796093022189, 35184367894509, 70368744175567

Offset: 1

N. J. A. Sloane, Jul 05 2022

If the binary expansion of a(n) has a 1 in the 2^i's bit (for any i >= 0) then A354169(2*n+1) must have a 0 in that bit.
A354169(2*n+1) is the smallest number not yet in A354169 which satisfies that condition (this follows at once from the definition of A354169).
This sequence bears the same relation to A354169 as A355057 does to A090252.

			Consider n=6. Then b(7) to b(12) are 32, 64, 12, 128, 256, 512. The bitwise OR of those 6 numbers is 1111101100_2 = 1004_10 = a(6). The bitwise complement of 1004_10 is 10011_2 = 19_10 = A354781(6), and A354169(6) = 17_10 = 10001_2.
On the other hand, for n=5, b(6) to b(10) are 16, 32, 64, 12, 128, whose bitwise OR is 11111100_2 = 252_10 = a(5). The bitwise complement of 252_10 is 3_10 = 11_2 = A354781(5). However, 3 has already appeared in A354169, and the smallest available number whose binary expansion is disjoint from 252_10 = 11111100_2 is 2^8 = 100000000_2 = 256_10 = 2^8 = A354169(5).

N. J. A. Sloane, Table of n, a(n) for n = 1..256

A bisection of A354757.
Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354781.
See also A090252, A355057.

A356803 a(n) = product of prohibited prime factors of A354790(n).

Original entry on oeis.org

1, 1, 2, 6, 15, 105, 385, 2310, 6006, 102102, 277134, 6374082, 16804398, 520936338, 3038795305, 66853496710, 190275336790, 7420738134810, 17897074325130, 769574195980590, 1903683537425670, 100895227483560510, 258818192240437830, 15787909726666707630, 36475515575402393490

Offset: 1

Michael De Vlieger, Sep 06 2022

nonn

Let s(n) = A354790(n), a squarefree number by definition. Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n. Therefore a(n) is the product of primes p that cannot divide s(n). a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. (After N. J. A. Sloane in A355057.)

Analogous to A355057.

			a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * s(2) / s((3-1)/2) = 1 * 2 / 1 = 2;
a(4) = a(3) * s(3) = 2 * 3 = 6;
a(5) = a(4) * s(4) / s((5-1)/2) = 6 * 5 / 2 = 15;
a(6) = a(5) * s(5) = 15 * 7 = 105;
a(7) = a(6) * s(6) / s((7-1)/2) = 105 * 11 / 3 = 385; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..585
Michael De Vlieger, Annotated plot of prime p | m at (n, pi(p)) for m = a(n) in blue and A354790(n) in red and n = 1..80.
Michael De Vlieger, Plot of prime p | m at (n, pi(p)) for m = a(n) in cyan and A354790(n) in red and n = 1..1024.

Cf. A354790, A355057.

Block[{s = Import["https://oeis.org/A354790/b354790.txt", "Data"][[1 ;; 26, -1]], ww, m = 1, t, w = 3, k = 3}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s] - 1}]][[-1, -1]] ]

a(n) = a(n-1) * s(n-1) / s((n-1)/2), where the last operation is only carried out iff (n-1)/2 is an integer.

cp's OEIS Frontend

A354765 a(n) is a binary encoded version of A355057(n).

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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.

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A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

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A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

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A356803 a(n) = product of prohibited prime factors of A354790(n).

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