A354169 -id:A354169 - cp's OEIS frontend

A355889 Concatenate the exponents of the powers of 2 in A354169(k) in increasing order, for k = 1, 2, 3, ...

0, 1, 2, 3, 0, 1, 4, 5, 6, 2, 3, 7, 8, 9, 0, 4, 10, 1, 5, 11, 12, 13, 2, 6, 14, 3, 7, 15, 16, 17, 8, 9, 18, 19, 20, 0, 10, 21, 1, 4, 22, 5, 11, 23, 24, 25, 12, 13, 26, 27, 28, 2, 14, 29, 3, 6, 30, 7, 15, 31, 32, 33, 16, 17, 34, 35, 36, 8, 18, 37, 9, 19, 38, 39, 40, 0, 20, 41, 10, 21, 42, 43, 44, 1, 22, 45, 4, 5, 46

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Jul 20 2022

nonn

It is conjectured that the Hamming weight of A354169(k) is always 0, 1, or 2. This is known to be true for at least the first 2^25 terms. (The present sequence is well-defined even if the conjecture is false.)
So this is a far more efficient way to present A354169 than by listing the decimal expansions.
The terms of A354169 that are pure powers of 2 appear in order, so it is obvious how to recover A354169 from this sequence.
This could be regarded as a table with (presumably) two columns, and could therefore have keyword "tabf", but that is not really appropriate, since basically it consists of the nonnegative integers with some interjections.

			A354169 begins 0, 1, 2, 4, 8, 3, 16, 32, 64, 12, 128, ... We ignore the initial 0, and then the binary expansions are 2^0, 2^1, 2^2, 2^3, 2^0+2^1, 2^4, 2^5, 2^6, 2^2+2^3, 2^7, ..., so the present sequence begins 0, 1, 2, 3, 0, 1, 4, 5, 6, 2, 3, 7, ...

Rémy Sigrist, Table of n, a(n) for n = 1..20000 (first 6585 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program

Cf. A354169, A355150, A354773, A354774, A354767, A354798, A354680.

A355701 a(n) = Product of prime(k+1) where k runs through the exponents of the positions 2^k of the 1-bits in A354169(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 6, 11, 13, 17, 35, 19, 23, 29, 22, 31, 39, 37, 41, 43, 85, 47, 133, 53, 59, 61, 667, 67, 71, 73, 62, 79, 33, 83, 481, 89, 97, 101, 1763, 103, 107, 109, 235, 113, 119, 127, 1007, 131, 137, 139, 3599, 149, 151, 157, 1541, 163, 2059, 167, 173, 179

Offset: 0

Michael De Vlieger, Jul 14 2022

nonn

Compactification of A354169. Offset matches A354169.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Annotated plot of prime factors p | a(n) at (n, pi(p)) for n = 0..50.

Cf. A005117, A090252, A354169.

nn = 58; r = c[] = 0; m = 1; Array[Set[{a[#], c[#]}, {#, #}] &, 2, 0]; Do[k = SelectFirst[Union@ Map[Total, Rest@ Subsets[2^Reverse[Length[#] - Position[#, 1][[All, 1]]] &@ IntegerDigits[2^(r + 2) - m - 1, 2]]], c[#] == 0 &]; Set[{a[n], c[k]}, {k, n}]; m += a[n]; If[And[IntegerQ[#], # > 0], m -= a[#]] &[n/2]; If[And[EvenQ[k], PrimePowerQ[k], k > 2^r], r++], {n, 2, nn}]; Table[Times @@ Map[Prime, 1 + Length[#] - Position[#, 1][[All, 1]]] &@ IntegerDigits[a[n], 2], {n, 0, nn}] (* _Michael De Vlieger, Jul 14 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

A354790 a(n) is the least positive squarefree number not already used that is coprime to the previous floor(n/2) terms.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 6, 13, 17, 19, 23, 29, 31, 35, 22, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 14, 97, 101, 103, 33, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 65, 233

Offset: 1

Michael De Vlieger and N. J. A. Sloane, Jul 17 2022

nonn

A version of the Two-Up sequence A090252 that is restricted to squarefree numbers.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Comments on A354790 regarding the possibility of composites with more than two factors.
Rémy Sigrist, Table of n, a(n) for n = 1..100000
Rémy Sigrist, PARI program with comments
Rémy Sigrist, C program (inspired by Russ Cox's Go program for A247665)
Michael De Vlieger, Compact annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n), n <= 1500. Color function indicates the number k > 1 of appearances of divisor p in the sequence. Diagram supports a proposition similar to Conjecture 3 in A090252 but regarding this sequence. Indices n connected in red appear in A355897.
Michael De Vlieger, Comprehensive annotated plot of prime p | A354790(n) at (n, pi(p)) for composite A354790(n), n <= 10^5. Color function indicates the number k > 1 of appearances of divisor p in the sequence.

Cf. A090252, A247665, A354169.
See A354791 and A354792 for the nonprime terms.
See A355895 for the even terms.

C
```
// See Links section.
```

# A354790 = Squarefree version of the Two-Up sequence A090252
# This produces 2*M terms in the array a
# Assumes b117 is a list of sufficiently many squarefree numbers A005117
# Test if u is relatively prime to all of a[i], i = i1..i2
perpq:=proc(u,i1,i2) local i; global a;
for i from i1 to i2 do if igcd(u,a[i])>1 then return(-1); fi; od: 1; end;
a:=Array(1..10000,-1);
hit:=Array(1..10000,-1); # 1 if i has appeared
a[1]:=1; a[2]:=2; hit[1]:=1; hit[2]:=1;
M:=100; M1 := 1000;
for p from 2 to M do
# step 1 want a[2p-1] relatively prime to a[p] ... a[2p-2]
sw1:=-1;
for j from 1 to M1 do
c:=b117[j];
if hit[c] = -1 and perpq(c,p,2*p-2) = 1 then a[2*p-1]:=c; hit[c]:=1; sw1:=1; break; fi;
od: # od j
if sw1 = -1 then error("no luck, step 1, p =",p ); fi;
# step 2 want a[2p] relatively prime to a[p] ... a[2p-1]
sw2:=-1;
for j from 1 to M1 do
c:=b117[j];
if hit[c] = -1 and perpq(c,p,2*p-1) = 1 then a[2*p]:=c; hit[c]:=1; sw2:=1; break; fi;
od: # od j
if sw2 = -1 then error("no luck, step 2, p =",p ); fi;
od: # od p
[seq(a[i],i=1..2*M)];

nn = 60; pp[] = 1; k = r = 1; c[] = False; a[1] = 1; Do[Set[m, SelectFirst[Union@ Append[Times @@ # & /@ Subsets[#, {2, Infinity}], Prime[r]] &[Prime@ Select[Range[If[k == 1, r, k + 1]], p[Prime[#]] < n &]], ! c[#] &]]; Set[a[n], m]; (c[m] = True; If[PrimeQ[m], r++]; If[n > 1, Map[(Set[p[#], 2 n]; pp[#]++) &, #]]) &@ FactorInteger[m][[All, 1]]; While[pp[Prime[k]] > 2, k++], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Sep 06 2022 *)

PARI
```
\\ See Links section.
```

from math import lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import factorint
def A354790_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1 and all(map(lambda n:n<=1,factorint(m).values())):
                yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A354790_list = list(islice(A354790_gen(),30)) # Chai Wah Wu, Jul 17 2022

More terms from Rémy Sigrist, Jul 17 2022

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Original entry on oeis.org

1, 3, 4, 12, 3, 19, 34, 64, 76, 136, 256, 768, 17, 1041, 50, 2080, 4096, 12288, 68, 16452, 200, 32896, 65536, 196608, 768, 262912, 524800, 1048576, 1049601, 2098176, 18, 4194322, 2096, 8390656, 16777216, 50331648, 12288, 67121152, 134225920, 268435456, 268451844, 536887296, 72, 1073741896, 32960, 2147516416, 4294967296, 12884901888

Offset: 1

N. J. A. Sloane, Jul 05 2022

			See A354780.

N. J. A. Sloane, Table of n, a(n) for n = 1..2400

Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354780.

A355893 Let A090252(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A090252(n); a(1)=0 by convention.

Original entry on oeis.org

0, 1, 10, 100, 2, 1000, 20, 10000, 100000, 1000000, 3, 10000000, 100000000, 200, 1010, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 4, 10000000000000000

Offset: 1

N. J. A. Sloane, Aug 23 2022

nonn

A090252 and A354169 are similar in many ways. This sequence and A355892 illustrate this.
This compressed format only make sense if all e_i are less than 10, that is, for n <= 24574.
It is believed that 6 does not appear in A090252, so 11 is missing from the present sequence.

			The initial terms of A090252 are:
1 -> 0
2 = 2^1 ->1
3 = 2^0 3^1 -> 10
5 = 2^0 3^0 5^1 -> 100
4 = 2^2 -> 2
7 = 2^0 3^0 5^0 7^1 -> 1000
9 = 2^0 3^2 -> 20
...
The terms, right-justified, for comparison with A355892, are:
.1 ...................................0
.2 ...................................1
.3 ..................................10
.4 .................................100
.5 ...................................2
.6 ................................1000
.7 ..................................20
.8 ...............................10000
.9 ..............................100000
10 .............................1000000
11 ...................................3
12 ............................10000000
13 ...........................100000000
14 .................................200
15 ................................1010
16 ..........................1000000000
17 .........................10000000000
18 ........................100000000000
19 .......................1000000000000
20 ......................10000000000000
21 .....................100000000000000
22 ....................1000000000000000
23 ...................................4
24 ...................10000000000000000
...

Michael De Vlieger, Table of n, a(n) for n = 1..1073

Cf. A054841, A090252, A354169.

See A354150 for indices of powers of 2 in A090252.

nn = 24, s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; nn, -1]]; f[n_] := If[n == 1, 0, Function[g, FromDigits@ Reverse@ ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ n]; Array[f[s[[#]]] &, nn] (* Michael De Vlieger, Aug 24 2022 *)

a(n) = A054841(A090252(n)). - Stefano Spezia, Aug 24 2022

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 4, 4, 12, 0, 3, 3, 19, 2, 34, 0, 64, 64, 76, 8, 136, 0, 256, 256, 768, 0, 17, 17, 1041, 16, 50, 32, 2080, 0, 4096, 4096, 12288, 0, 68, 68, 16452, 64, 200, 128, 32896, 0, 65536, 65536, 196608, 0, 768, 768, 262912, 512, 524800, 0, 1048576, 1048576, 1049601, 1024, 2098176, 0, 18, 18, 4194322, 16, 2096, 2048, 8390656, 0, 16777216

Offset: 1

N. J. A. Sloane, Jul 08 2022

Has the same relation to A354757 as A354781 does to A354780.

The offset is 1, to avoid having to define a(0).

			A354757(5) = 12 = 1100_2, so a(5) = 11_2 = 3.
A354757(6) = 15 = 1111_2, so a(6) = 0.
A354757(7) = 27 = 11011_2, so a(7) = 100_2 = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..4800

Cf. A354169, A354757, A354780, A354781.

See A354793 for Hamming weight of a(n).

Added comment and examples. - N. J. A. Sloane, Aug 02 2022

A354150 Index of 2^n in A090252, or -1 if 2^n does not appear.

Original entry on oeis.org

1, 2, 5, 11, 23, 95, 191, 383, 1535, 6143, 24575, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367

Offset: 0

N. J. A. Sloane, May 26 2022

It is conjectured (see A090252) that the indices of even terms in A090252 are {3*2^i-1, i >= 0} (A083329), so the positive terms of the present sequence should be a subsequence of A083329.

For n = 1,...,18, the terms are 3*2^k - 1 for k = 0,1,2,3,5,6,7,9,11,13,14,16,18,20,22,24,26,28. Will the formula a(n) = 3*2^(2*n-8)-1 hold for all n >= 11? This appears to be yet another example of the influence of A029744 on A090252 and A354691.- N. J. A. Sloane, Aug 24 2022

Cf. A090252, A083329, A354148, A354149, A354255 (even terms in A090252).

A354793 Hamming weight of A354783(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2

Offset: 1

N. J. A. Sloane, Jul 19 2022

nonn

Conjecture: This sequence appears to have a simple structure. Encode it by making the following substitutions, in this order:
Replace the initial 28 terms 0011201120223120113120112022 by S (as usual, the start is irregular), then map:
  3 1 3 -> 7
  3 1 2 -> 6
  1 2 0 1 1 2 0 2 2 -> 9
  0 1 1 -> 2
  0 2 2 -> 4
Then it appears that the encoded sequence is the concatenation of the following blocks:
  S
  79
  79(6264)^1
  79(6264)^1
  79(6264)^3
  79(6264)^3
  79(6264)^15
  79(6264)^15
  79(6264)^31
  79(6264)^31
  79(6264)^63
  79(6264)^63
  79(6264)^127
  79(6264)^127
  ...
This is probably not the most efficient encoding, but I was happy to find any one that revealed the structure.
From Michel Dekking, Jul 23 2022: (Start)
The following is another way to present the conjecture above, which shows the close connection with sequence A355150.
Conjecture: It appears that this sequence is almost a periodic sequence, with period 12. Let x:=A354789.
If n > 28, n == 5 (mod 12) is not an element of x then (written as words)
      a(n)a(n+1)...a(n+11) = 312011312022.
If n > 28, n == 5 (mod 12) is an element of x then
      a(n)a(n+1)...a(n+11) = 313120112022.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..4900

Cf. A354169, A354757, A354783, A355150.

A355714 Numbers k > 0 such that A090252(A355176(k)) does not equal prime(k)^2.

Original entry on oeis.org

1, 2, 16, 26, 32, 35, 40, 59, 60, 69, 92, 105, 110, 112, 113, 137, 167, 169, 178, 185, 186, 188, 207, 210, 260, 261, 274, 287, 289, 342, 344, 346, 356, 357, 359, 361, 362, 363, 391, 412, 417, 434, 457, 477, 478, 479, 480, 481, 492, 547, 563, 598, 663, 666, 671

Offset: 1

Thomas Scheuerle, Jul 15 2022

nonn

In contrast to A354169 (a set theoretic analog of A090252) where we observe many as yet unproved regularities, in A090252 the situation appears to be more complicated. This sequence is intended to help spot these irregularities and perhaps lead to further rules, which probably have no analogy in A354169.
In most cases A090252(A355176(n)) equals prime(a(n))*p where p is prime. Surprisingly p is in many cases greater than prime(a(n)).
Is this sequence infinite?

Cf. A000040, A001248, A090252, A354169, A355176.

A090252(A355176(a(n))) <> A000040(a(n))^2.
A000040(a(n)) divides A090252(A355176(a(n))).
A000040(a(n)) divides A090252(2*A355176(a(n))+1).

More terms from Jinyuan Wang, Jul 15 2022

cp's OEIS Frontend

A355889 Concatenate the exponents of the powers of 2 in A354169(k) in increasing order, for k = 1, 2, 3, ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A355701 a(n) = Product of prime(k+1) where k runs through the exponents of the positions 2^k of the 1-bits in A354169(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A354790 a(n) is the least positive squarefree number not already used that is coprime to the previous floor(n/2) terms.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

C

Maple

Mathematica

PARI

Python

Extensions

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Views

Author

Keywords

Examples

Links

Crossrefs

A355893 Let A090252(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A090252(n); a(1)=0 by convention.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A354150 Index of 2^n in A090252, or -1 if 2^n does not appear.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A354793 Hamming weight of A354783(n).

Views