A090252 -id:A090252 - 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

A355014 Index of A355012(n) in A246547.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 11, 7, 14, 12, 17, 10, 19, 21, 9, 24, 18, 25, 13, 28, 29, 30, 34, 15, 36, 39, 41, 43, 44, 27, 45, 48, 38, 16, 53, 54, 55, 22, 57, 49, 51, 52, 33, 71, 76, 77, 60, 63, 64, 65, 67, 23, 68, 70, 88, 20, 89, 90, 72, 75, 96, 97, 98, 42, 79, 80, 81

Offset: 1

Michael De Vlieger, Jun 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1009

Cf. A090252, A246547, A355012, A355013.

With[{s = Map[ToExpression@ StringSplit[#][[-1]] &, Import["https://oeis.org/A246547/b246547.txt", "Data"][[2 ;; -1]]]}, Map[FirstPosition[s, #][[1]] &, Select[Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 8000, -1]], CompositeQ[#] && PrimePowerQ[#] &] ]]

A355892 a(n) = binary expansion of A354169(n).

Original entry on oeis.org

0, 1, 10, 100, 1000, 11, 10000, 100000, 1000000, 1100, 10000000, 100000000, 1000000000, 10001, 10000000000, 100010, 100000000000, 1000000000000, 10000000000000, 1000100, 100000000000000, 10001000, 1000000000000000, 10000000000000000, 100000000000000000, 1100000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

N. J. A. Sloane, Aug 23 2022

nonn

			The terms, right-justified, for comparison with A355893.
...................................0
...................................1
..................................10
.................................100
................................1000
..................................11
...............................10000
..............................100000
.............................1000000
................................1100
............................10000000
...........................100000000
..........................1000000000
...............................10001
.........................10000000000
..............................100010
........................100000000000
.......................1000000000000
......................10000000000000
.............................1000100
.....................100000000000000
............................10001000
....................1000000000000000
...................10000000000000000
..................100000000000000000
..........................1100000000
.................1000000000000000000
................10000000000000000000
...............100000000000000000000
.........................10000000001
..............1000000000000000000000
...............................10010
.............10000000000000000000000
........................100000100000
............100000000000000000000000
...........1000000000000000000000000
..........10000000000000000000000000
......................11000000000000
.........100000000000000000000000000
........1000000000000000000000000000
.......10000000000000000000000000000
.....................100000000000100
......100000000000000000000000000000
.............................1001000
.....1000000000000000000000000000000
....................1000000010000000
....10000000000000000000000000000000
...100000000000000000000000000000000
..1000000000000000000000000000000000
..................110000000000000000

N. J. A. Sloane, Table of n, a(n) for n = 0..999

Cf. A354169, A090252, A355893.

A355065 Lexicographically earliest sequence of distinct positive integers such that if m and n are distinct and not coprime, then a(n) does not belong to the interval ceiling(a(m)/2)..2*a(m).

Original entry on oeis.org

1, 2, 3, 5, 4, 11, 6, 23, 24, 47, 7, 95, 8, 191, 192, 383, 9, 767, 10, 1535, 1536, 3071, 12, 6143, 3072, 12287, 12288, 24575, 13, 49151, 14, 98303, 98304, 196607, 98305, 393215, 15, 786431, 786432, 1572863, 16, 3145727, 17, 6291455, 6291456, 12582911, 18

Offset: 1

Rémy Sigrist, Jun 17 2022

nonn

This sequence is a permutation of the nonnegative integers (when n is prime, a(n) is the least value not yet in the sequence).
The inverse sequence (A355066) has similarities with the Two-Up sequence (A090252) as A355066(n) is coprime to the next n terms (and to the floor(n/2) previous terms).
Note that the relation "u does not belong to the interval ceiling(v/2)..2*v" is symmetrical (for u, v > 0).

			The first terms, alongside the forbidden values, are:
  n   a(n)  Forbidden values
  --  ----  ---------------------------------------------------
   1     1  None
   2     2  None
   3     3  None
   4     5  1..4 (from m=2)
   5     4  None
   6    11  1..4 (from m=2), 2..6 (from m=3), 3..10 (from m=4)
   7     6  None
   8    23  1..4 (from m=2), 3..10 (from m=4), 6..22 (from m=6)
   9    24  2..6 (from m=3), 6..22 (from m=6)
  10    47  1..4 (from m=2), 3..10 (from m=4), 2..8 (from m=5),
            6..22 (from m=6), 12..46 (from m=8)
  11     7  None

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A090252, A355066 (inverse).

PARI
```
See Links section.
```

A355066 Inverse permutation to A355065.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 13, 17, 19, 6, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 8, 9, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 77, 137, 139, 149, 151, 157, 163, 167, 10, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Rémy Sigrist, Jun 17 2022

nonn

This sequence has similarities with the Two-Up sequence (A090252) as a(n) is coprime to the next n terms (and to the floor(n/2) previous terms).

Prime numbers appear in their natural order.

			A355065(77) = 39, so a(39) = 77.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A090252, A355065.

PARI
```
See Links section.
```

A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 29, 27, 20, 31, 37, 12, 41, 43, 14, 15, 47, 22, 53, 39, 32, 25, 49, 18, 59, 61, 34, 45, 67, 38, 71, 33, 28, 65, 73, 24, 79, 83, 46, 75, 89, 56, 97, 81, 44, 85, 101, 26, 87, 103, 62, 35, 57, 64, 107

Offset: 1

Sergio Pimentel, May 14 2021

nonn

It is not clear if every positive integer is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A344307

Cf. A064413, A084937, A090252, A103683, A121216, A249064.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, May 29 2021

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 *)

A355894 Let A354790(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 A354790(n); a(1)=0 by convention.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 11, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 1100, 10001, 100000000000, 100010, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000

Offset: 0

N. J. A. Sloane, Aug 25 2022

nonn

The terms of A354790 are squarefree, so here the exponents e_i are 0 or 1.

This bears the same relation to A354790 as A355893 does to A090252.

			The terms, right-justified, for comparison with A355892 and A355893, are:
   1 ...................................0
   2 ...................................1
   3 ..................................10
   4 .................................100
   5 ................................1000
   6 ...............................10000
   7 ..................................11
   8 ..............................100000
   9 .............................1000000
  10 ............................10000000
  11 ...........................100000000
  12 ..........................1000000000
  13 .........................10000000000
  14 ................................1100
  15 ...............................10001
  16 ........................100000000000
  17 ..............................100010
  18 .......................1000000000000
  19 ......................10000000000000
  20 .....................100000000000000
  21 ....................1000000000000000
  22 ...................10000000000000000
  23 ..................100000000000000000
  24 .................1000000000000000000
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..1051

Cf. A090252, A354790, A355893.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A355014 Index of A355012(n) in A246547.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A355892 a(n) = binary expansion of A354169(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A355065 Lexicographically earliest sequence of distinct positive integers such that if m and n are distinct and not coprime, then a(n) does not belong to the interval ceiling(a(m)/2)..2*a(m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A355066 Inverse permutation to A355065.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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

A355894 Let A354790(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 A354790(n); a(1)=0 by convention.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs