A353715 -id:A353715 - cp's OEIS frontend

A353724 a(n) = exponent of highest power of 2 that divides A353715(n).

0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 3, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 2, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 3, 0, 0, 2, 0, 0, 1, 5, 0, 0, 4, 0, 0, 2, 1, 0, 0, 3, 0, 0, 3, 6, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 0

Offset: 0

N. J. A. Sloane, May 11 2022

nonn

			A353715(3) = 12 = 2^2*3, so a(3) = 2. A353715(11) = 104 = 2^3*13, so a(11) = 3.

Cf. A353709, A353715, A353725, A353726.

from itertools import count, islice
def A353724_gen(): # generator of terms
    s, a, b, c, ab = {0,1}, 0, 1, 2, 1
    yield 0
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                yield len(t := bin(b+n))-len(t.rstrip('0'))
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353724_list = list(islice(A353724_gen(),30)) # Chai Wah Wu, May 11 2022

A353727 Index in A353715 of the first term divisible by 2^n and no higher power of 2, or -1 if no such term exists.

Original entry on oeis.org

0, 2, 3, 11, 54, 74, 88, 183, 20334, 30938, 21247, 90575, 3913, 124845, 2643790, 5828721, 2469947, 4005550, 19917707

Offset: 0

Walter Trump, May 11 2022

			Table showing initial values of n (column 1) and a(n) (column 3).
The central column shows the corresponding entry of A353715 written in base 2.
The entries in column 2 end in exactly n zeros.
   n                    A353715(a(n))       a(n)
   0                                1         0
   1                              110         2
   2                             1100         3
   3                          1101000        11
   4                         11110000        54
   5                       1111100000        74
   6                      11101000000        88
   7                     110110000000       183
   8              1111110101100000000     20334
   9             10111110101000000000     30938
  10              1111111110000000000     21247
  11           1001111111100000000000     90575
  12                11111000000000000      3913
  13          10011111110000000000000    124845
  14      111110111110100000000000000   2643790
  15    10011111111111000000000000000   5828721
  16      111111111110000000000000000   2469947
  17     1111111101100000000000000000   4005550
  18  1011111111111000000000000000000  19917707

Cf. A353709, A353715, A353724, A353725, A353726.

A353709 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6, 40, 64, 5, 10, 48, 65, 14, 128, 33, 18, 68, 9, 34, 20, 72, 35, 132, 24, 66, 36, 25, 130, 96, 13, 144, 98, 256, 21, 42, 192, 257, 22, 104, 129, 258, 28, 97, 384, 26, 37, 320, 136, 7, 80, 160, 11, 84, 288, 131, 76, 272, 161, 70, 264, 49, 134, 328, 512, 19, 44, 448, 513, 30, 224, 768, 15, 112, 640, 259, 52, 200, 514, 53

Offset: 0

N. J. A. Sloane, May 06 2022

A set-theory analog of A084937.

Conjecture: This is a permutation of the nonnegative numbers.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Walter Trump, Table of n, a(n) for n = 0..10^6
Walter Trump, Log-log plot of first 2^24 terms

Cf. A084937 (number theory analog), A109812, A121216, A353405 (powers of 2), A353708, A353710, A353715 and A353716 (a(n)+a(n+1)), A353717 (inverse), A353718, A353719 (primes), A353720 and A353721 (Records).

For the numbers that are the slowest to appear see A353723 and A353722.

read(transforms) : # ANDnos def'd here
A353709 := proc(n)
option remember;
local c, i, known ;
if n <= 2 then
n;
else
for c from 1 do
known := false ;
for i from 1 to n-1 do
if procname(i) = c then
known := true;
break ;
end if;
end do:
if not known and ANDnos(c, procname(n-2)) = 0 and ANDnos(c, procname(n-1)) = 0 then
return c;
end if;
end do:
end if;
end proc: # Following R. J. Mathar's program for A109812.
[seq(A353709(n), n=0..256)] ;
# second Maple program:
b:= proc() false end: t:= 2:
a:= proc(n) option remember; global t; local k; if n<2 then n
      else for k from t while b(k) or Bits[And](k, a(n-2))>0
      or Bits[And](k, a(n-1))>0 do od; b(k):=true;
      while b(t) do t:=t+1 od; k fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 06 2022

nn = 83; c[] = -1; a[0] = c[0] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[a[n - 1], k] == 0, BitAnd[a[n - 2], k] == 0], k++]; Set[{a[n], c[k]}, {k, n}]; If[k == u, While[c[u] > -1, u++]], {n, 2, nn}]; Array[a, nn+1, 0] (* _Michael De Vlieger, May 06 2022 *)

from itertools import count, islice
def A353709_gen(): # generator of terms
    s, a, b, c, ab = {0,1}, 0, 1, 2, 1
    yield from (0,1)
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                yield n
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353709_list = list(islice(A353709_gen(),20)) # Chai Wah Wu, May 07 2022

A353725 Records in A353724.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 16, 17, 18

Offset: 1

N. J. A. Sloane, May 11 2022

Motivated by a comment in A353715.

			Table from _Walter Trump_, May 11 2022, showing initial terms of A353725 (column 1) and A353726 (column 3). The central column shows the corresponding entry of A353715 written in base 2.
   0                                1         0
   1                              110         2
   2                             1100         3
   3                          1101000        11
   4                         11110000        54
   5                       1111100000        74
   6                      11101000000        88
   7                     110110000000       183
  12                11111000000000000      3913
  13          10011111110000000000000    124845
  16      111111111110000000000000000   2469947
  17     1111111101100000000000000000   4005550
  18  1011111111111000000000000000000  19917707

Cf. A353709, A353715, A353724, A353726, A353727.

a(10)-a(13) from Walter Trump, May 11 2022

A353726 Indices of records in A353724.

Original entry on oeis.org

0, 2, 3, 11, 54, 74, 88, 183, 3913, 124845, 2469947, 4005550, 19917707

Offset: 1

N. J. A. Sloane, May 11 2022

Cf. A353709, A353715, A353724, A353725, A353727.

a(10)-a(13) from Walter Trump, May 11 2022

A353716 The values of b(k)+b(k+1), k >= 0, sorted into increasing order, where b is A353709.

Original entry on oeis.org

1, 3, 6, 11, 12, 15, 19, 23, 28, 43, 44, 46, 49, 51, 54, 58, 61, 63, 69, 77, 79, 86, 87, 90, 92, 95, 102, 104, 107, 109, 113, 123, 125, 126, 127, 142, 143, 155, 156, 157, 161, 167, 171, 183, 187, 191, 207, 221, 226, 231, 233, 234, 239, 240, 242, 252, 253, 254, 255, 277, 279, 286, 311, 313, 319, 333, 334, 348, 351, 354, 357, 365, 372, 383, 387

Offset: 1

N. J. A. Sloane, May 09 2022

nonn

The sequence [A353709(k)+A353709(k+1) for k >= 0] (unsorted) is A353715.

Cf. A353709, A353715.

cp's OEIS Frontend

A353724 a(n) = exponent of highest power of 2 that divides A353715(n).

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A353727 Index in A353715 of the first term divisible by 2^n and no higher power of 2, or -1 if no such term exists.

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A353709 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.

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A353725 Records in A353724.

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A353726 Indices of records in A353724.

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A353716 The values of b(k)+b(k+1), k >= 0, sorted into increasing order, where b is A353709.

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