A353709 -id:A353709 - 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

A353717 a(n) = index of n in A353709, or -1 if n does not appear there.

Original entry on oeis.org

0, 1, 2, 5, 3, 13, 10, 53, 4, 22, 14, 56, 7, 34, 17, 76, 6, 9, 20, 69, 24, 38, 42, 86, 28, 31, 49, 90, 46, 185, 73, 245, 8, 19, 23, 26, 30, 50, 94, 322, 11, 102, 39, 105, 70, 109, 252, 549, 15, 65, 98, 117, 80, 83, 124, 1047, 113, 225, 242, 692, 249, 1209, 553, 1647, 12, 16, 29, 114, 21, 137, 63, 329, 25, 99, 133, 312, 60, 140, 339, 1007, 54, 175, 148, 189, 57, 238

Offset: 0

N. J. A. Sloane, May 09 2022

nonn

If, as conjectured, A353709 is a permutation of the nonnegative integers, then this is the inverse permutation.

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Rémy Sigrist, C++ program

Cf. A353709.

b:= proc() false end: t:= 2:
g:= 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, g(n-2))>0
      or Bits[And](k, g(n-1))>0 do od; b(k):=true;
      while b(t) do t:=t+1 od; k fi
    end:
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= g(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..85);  # Alois P. Heinz, May 09 2022

A353710 Smallest missing number when A353709(n) is being calculated.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 27, 27, 27, 27, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29

Offset: 0

N. J. A. Sloane, May 06 2022

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, PARI program

Cf. A084937, A353709.

PARI
```
See Links section.
```

from itertools import count, islice
def A353710_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 c
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353710_list = list(islice(A353710_gen(),20)) # Chai Wah Wu, May 10 2022

A353720 Records in A353709.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 40, 64, 65, 128, 132, 144, 256, 257, 258, 384, 512, 513, 768, 832, 1024, 1025, 1026, 1028, 1032, 1152, 1280, 1408, 2048, 2049, 2052, 2056, 2304, 3072, 3073, 3074, 3080, 3104, 4096, 4098, 4100, 4608, 4864, 5120, 5121, 5124, 6144, 8192, 8193, 8224, 8256

Offset: 1

N. J. A. Sloane, May 10 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..800 (first 137 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program

Cf. A353709, A353721.

A353721 Indices of records in A353709.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 11, 12, 16, 18, 27, 35, 37, 41, 45, 48, 68, 72, 75, 88, 89, 93, 108, 112, 119, 122, 180, 183, 184, 214, 220, 231, 237, 244, 305, 308, 321, 328, 338, 400, 404, 410, 423, 435, 442, 459, 490, 548, 552, 669, 753, 759, 799, 931, 1002, 1006, 1009, 1029, 1096, 1174, 1228, 1399

Offset: 1

N. J. A. Sloane, May 10 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..800 (first 137 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program

Cf. A353709, A353720.

Data corrected by Rémy Sigrist, May 11 2022

A353405 Index of 2^n in A353709.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 18, 37, 68, 89, 184, 338, 548, 1029, 2141, 3914, 7121, 11830, 22923, 38692, 75029, 124846, 223140, 419105, 807096, 1385673, 2636205, 4649883, 8759535, 16901645, 29744020, 56292997, 105932907

Offset: 0

Rémy Sigrist, May 07 2022

			A353709(1029) = 16384 = 2^14, so a(14) = 1029.

Rémy Sigrist, C++ program

Cf. A305370, A353709.

A353719 Index of prime(n) in A353709, or -1 if prime(n) does not appear in A353709.

Original entry on oeis.org

2, 5, 13, 53, 56, 34, 9, 69, 86, 185, 245, 50, 102, 105, 549, 83, 692, 1209, 114, 329, 99, 1007, 189, 235, 47, 319, 542, 740, 724, 232, 5257, 59, 159, 373, 480, 1100, 1371, 476, 1141, 1138, 1044, 498, 18890, 156, 363, 867, 929, 7890, 1041, 925, 564, 12929, 682

Offset: 1

N. J. A. Sloane, May 09 2022

Prime(n) refers to the n-th term in the sequence of primes, not the n-th prime in A353709.

			A353709 has offset 0 and begins 0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6, 40, 64, 5, 10, ..., so a(1) = 2 (from A353709(2) = 2), and a(7) = 9 (from A353709(9) = 17 = prime(7)).

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, C++ program

Cf. A353709, A353717.

More terms from Rémy Sigrist, May 09 2022

A354040 a(n) is the least k such that A353709(k) has Hamming weight n.

Original entry on oeis.org

0, 1, 5, 17, 73, 245, 1037, 4624, 11829, 30940, 116701, 374756, 1205149, 2643792, 7516934, 27309591, 102243621

Offset: 0

Rémy Sigrist, May 15 2022

This sequence is strictly increasing.

			For n = 3:
- the first terms of A353709, alongside their Hamming weight, are:
  k             | 0, 1, 2, 3, 4, 5, 6,  7,  8,  9,  10, 11, 12, 13, 14, 15, 16, 17
  A353709(k)    | 0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6,  40, 64, 5,  10, 48, 65, 14
  h(A353709(k)) | 0, 1, 1, 1, 1, 2, 1,  2,  1,  2,  2,  2,  1,  2,  2,  2,  2,  3
- the first term of A353709 with Hamming weight 3 is A353709(17) = 14,
- so a(3) = 17.

Rémy Sigrist, C++ program

Cf. A000120, A353709, A354041.

A354041 a(n) is the earliest term of A353709 with Hamming weight n.

Original entry on oeis.org

0, 1, 3, 14, 30, 31, 252, 508, 1020, 511, 1535, 2047, 4095, 8191, 16383, 49151, 65535

Offset: 0

Rémy Sigrist, May 15 2022

This sequence is not strictly increasing.

			For n = 3:
- the first terms of A353709, alongside their Hamming weight, are:
  k             | 0, 1, 2, 3, 4, 5, 6,  7,  8,  9,  10, 11, 12, 13, 14, 15, 16, 17
  A353709(k)    | 0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6,  40, 64, 5,  10, 48, 65, 14
  h(A353709(k)) | 0, 1, 1, 1, 1, 2, 1,  2,  1,  2,  2,  2,  1,  2,  2,  2,  2,  3
- the first term of A353709 with Hamming weight 3 is A353709(17) = 14,
- so a(3) = 14.

Rémy Sigrist, C++ program

Cf. A000120, A353709, A354040.

a(n) = A353709(A354040(n)).

A352690 Index of 2^n-1 in A353709.

Original entry on oeis.org

0, 1, 5, 53, 76, 245, 1647, 5257, 20336, 30940, 124837, 374756, 1205149, 2643792, 7516934, 29855796, 102243621

Offset: 0

Rémy Sigrist, May 15 2022

This sequence is strictly increasing.

			A353709(1647) = 63 = 2^6 - 1, so a(6) = 1647.

Rémy Sigrist, C++ program

Cf. A000225, A353709, A353405.

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

A353717 a(n) = index of n in A353709, or -1 if n does not appear there.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A353710 Smallest missing number when A353709(n) is being calculated.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

A353720 Records in A353709.

Views

Author

Keywords

Links

Crossrefs

A353721 Indices of records in A353709.

Views

Author

Keywords

Links

Crossrefs

Extensions

A353405 Index of 2^n in A353709.

Views

Author

Keywords

Examples

Links

Crossrefs

A353719 Index of prime(n) in A353709, or -1 if prime(n) does not appear in A353709.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A354040 a(n) is the least k such that A353709(k) has Hamming weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A354041 a(n) is the earliest term of A353709 with Hamming weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A352690 Index of 2^n-1 in A353709.

Views

Author

Keywords