A353726 -id:A353726 - cp's OEIS frontend

A353715 a(n) = b(n)+b(n+1), where b is A353709.

1, 3, 6, 12, 11, 19, 28, 44, 49, 23, 46, 104, 69, 15, 58, 113, 79, 142, 161, 51, 86, 77, 43, 54, 92, 107, 167, 156, 90, 102, 61, 155, 226, 109, 157, 242, 354, 277, 63, 234, 449, 279, 126, 233, 387, 286, 125, 481, 410, 63, 357, 456, 143, 87, 240, 171, 95, 372, 419, 207, 348, 433, 231, 334, 313, 183, 462, 840, 531, 63, 492, 961, 543, 254, 992, 783, 127

Offset: 0

N. J. A. Sloane, May 09 2022

Created in an attempt to show that every number appears in A353709. For example, if one could show that the present sequence had a subsequence which was divisible by ever-increasing powers of 2, the desired result would follow. See A353724, A353725, A353726, A353727 for more about this topic.

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
Walter Trump, Log-log plot of first 2^24 terms. Green dots (which overwrite the black dots) indicate terms a(n) which are divisible by 64.

Cf. A353709, A353724, A353725, A353726, A353727.

g:= proc() false end: t:= 2:
b:= proc(n) option remember; global t; local k; if n<2 then n
      else for k from t while g(k) or Bits[And](k, b(n-2))>0
      or Bits[And](k, b(n-1))>0 do od; g(k):=true;
      while g(t) do t:=t+1 od; k fi
    end:
a:= n-> b(n)+b(n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, May 09 2022

g[_] = False ; t = 2;
b[n_] := b[n] = Module[{k}, If[n < 2, n,
   For[k = t, g[k] || BitAnd[k, b[n-2]] > 0 ||
   BitAnd[k, b[n-1]] > 0, k++]; g[k] = True;
   While[g[t], t = t+1]; k]];
a[n_] := b[n] + b[n+1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 07 2022, after Alois P. Heinz *)

from itertools import count, islice
def A353715_gen(): # generator of terms
    s, a, b, c, ab = {0,1}, 0, 1, 2, 1
    yield 1
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                yield b+n
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353715_list = list(islice(A353715_gen(),30)) # Chai Wah Wu, May 11 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

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A353715 a(n) = b(n)+b(n+1), where b is A353709.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A353725 Records in A353724.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Examples

Crossrefs