A181391 -id:A181391 - cp's OEIS frontend

A358258 First n-bit number to appear in Van Eck's sequence (A181391).

0, 2, 6, 9, 17, 42, 92, 131, 307, 650, 1024, 2238, 4164, 8226, 17384, 33197, 67167, 133549, 269119, 525974, 1055175, 2111641, 4213053, 8444257, 16783217, 33601813, 67405064, 134239260, 268711604, 538400994, 1076155844, 2152693259, 4299075300, 8594396933, 17203509931

Offset: 1

Michael De Vlieger, Nov 05 2022

Binary version of A358168.

			First terms written in binary, substituting "." for 0 to enhance the pattern of 1's.
   n      a(n)                   a(n)_2
  -------------------------------------
   1        0                         .
   2        2                        1.
   3        6                       11.
   4        9                      1..1
   5       17                     1...1
   6       42                    1.1.1.
   7       92                   1.111..
   8      131                  1.....11
   9      307                 1..11..11
  10      650                1.1...1.1.
  11     1024               1..........
  12     2238              1...1.11111.
  13     4164             1.....1...1..
  14     8226            1.......1...1.
  15    17384           1....11111.1...
  16    33197          1......11.1.11.1
  17    67167         1.....11..1.11111
  18   133549        1.....1..11.1.11.1
  19   269119       1.....11.11..111111
  20   525974      1........11.1..1.11.
  21  1055175     1.......11..111...111
  22  2111641    1.......111...1..11..1
  23  4213053   1.......1..1..1..1111.1
  24  8444257  1.......11.11..1.11....1

Cf. A181391, A358168, A358180, A358259.

nn = 2^20; q[] = False; q[0] = True; a[] = 0; c[_] = -1; c[0] = 2; m = 1; {0}~Join~Rest@ Reap[Do[j = c[m]; k = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#], Sow[k]; q[#] = True] & @ IntegerLength[k, 2], {n, 3, nn}] ][[-1, -1]]

from itertools import count
def A358258(n):
    b, bdict, k = 0, {0:(1,)},1< 1 else 0
    for m in count(2):
        if b >= k:
            return b
        if len(l := bdict[b]) > 1:
            b = m-1-l[-2]
            if b in bdict:
                bdict[b] = (bdict[b][-1],m)
            else:
                bdict[b] = (m,)
        else:
            b = 0
            bdict[0] = (bdict[0][-1],m) # Chai Wah Wu, Nov 06 2022

a(30)-a(34) from Chai Wah Wu, Nov 06 2022

a(35) from Martin Ehrenstein, Nov 07 2022

A358259 Positions of the first n-bit number to appear in Van Eck's sequence (A181391).

Original entry on oeis.org

1, 5, 10, 24, 41, 52, 152, 162, 364, 726, 1150, 2451, 4626, 9847, 18131, 36016, 71709, 143848, 276769, 551730, 1086371, 2158296, 4297353, 8607525, 17159741, 34152001, 68194361, 136211839, 271350906, 541199486, 1084811069, 2165421369, 4331203801, 8643518017, 17303787585

Offset: 1

Michael De Vlieger, Nov 05 2022

Binary version of the concept behind A358180.

			First terms written in binary, substituting "." for 0 to enhance the pattern of 1's.
   n      a(n)                   a(n)_2
  -------------------------------------
   1        1                         1
   2        5                       1.1
   3       10                      1.1.
   4       24                     11...
   5       41                    1.1..1
   6       52                    11.1..
   7      152                  1..11...
   8      162                  1.1...1.
   9      364                 1.11.11..
  10      726                1.11.1.11.
  11     1150               1...111111.
  12     2451              1..11..1..11
  13     4626             1..1....1..1.
  14     9847            1..11..111.111
  15    18131           1...11.11.1..11
  16    36016          1...11..1.11....
  17    71709         1...11......111.1
  18   143848        1...11...1111.1...
  19   276769       1....111..1..1....1
  20   551730      1....11.1.11..11..1.
  21  1086371     1....1..1..111.1...11
  22  2158296    1.....111.111.11.11...
  23  4297353   1.....11..1..1.1...1..1
  24  8607525  1.....11.1.1.111..1..1.1
  etc.

Cf. A181391, A358168, A358180, A358258.

nn = 2^20; q[] = False; q[0] = True; a[] = 0; c[_] = -1; c[0] = 2; m = 1; {1}~Join~Rest@ Reap[Do[j = c[m]; k = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#], Sow[n]; q[#] = True] & @ IntegerLength[k, 2], {n, 3, nn}] ][[-1, -1]]

from itertools import count
def A358259(n):
    b, bdict, k = 0, {0:(1,)},1< 1 else 0
    for m in count(2):
        if b >= k:
            return m-1
        if len(l := bdict[b]) > 1:
            b = m-1-l[-2]
            if b in bdict:
                bdict[b] = (bdict[b][-1],m)
            else:
                bdict[b] = (m,)
        else:
            b = 0
            bdict[0] = (bdict[0][-1],m) # Chai Wah Wu, Nov 06 2022

a(30)-a(34) from Chai Wah Wu, Nov 06 2022

a(35) from Martin Ehrenstein, Nov 07 2022

A171912 Van Eck sequence (cf. A181391) starting with 2.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 5, 0, 3, 0, 2, 5, 5, 1, 10, 0, 6, 0, 2, 8, 0, 3, 13, 0, 3, 3, 1, 13, 5, 16, 0, 7, 0, 2, 15, 0, 3, 11, 0, 3, 3, 1, 15, 8, 24, 0, 7, 15, 5, 20, 0, 5, 3, 12, 0, 4, 0, 2, 24, 14, 0, 4, 6, 46, 0, 4, 4, 1, 26, 0, 5, 19, 0, 3, 21, 0, 3, 3, 1, 11, 42, 0, 6, 20, 34, 0, 4, 20, 4

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty. - M. F. Hasler, Jun 12 2019

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

t = {2};
Do[
d = Quiet[Check[Position[t, Last[t]][[-2]][[1]], 0]];
If[d == 0, x = 0, x = Length[t] - d];
AppendTo[t, x], 100]
t  (* Horst H. Manninger, Aug 30 2020 *)

A171912_vec(N, a=2, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 11 2019

from itertools import count, islice
def A171912gen(): # generator of terms
    b, bdict = 2, {2:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A171912_list = list(islice(A171912gen(),20)) # Chai Wah Wu, Dec 21 2021

Name edited and cross-references added by M. F. Hasler, Jun 12 2019

A171913 Van Eck sequence (cf. A181391) starting with a(1) = 3.

Original entry on oeis.org

3, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 20, 0, 4, 6, 9, 0, 4, 4, 1, 20, 9, 6, 8, 0, 8, 2, 22, 0, 4, 11, 0, 3, 22, 6, 12, 0, 5, 28, 0, 3, 8, 16, 0, 4, 15, 0, 3, 7, 0, 3, 3, 1, 33, 0, 5, 18, 0, 3, 7, 11, 30, 0, 5, 8, 23, 0, 4, 23, 3, 11, 10, 0, 6, 39, 0, 3, 7, 18, 22

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty, i.e., a(n) does not appear earlier in the sequence. - M. F. Hasler, Jun 12 2019

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

t = {3};
Do[
d = Quiet[Check[Position[t, Last[t]][[-2]][[1]], 0]];
If[d == 0, x = 0, x = Length[t] - d];
AppendTo[t, x], 100]
t  (* Horst H. Manninger, Sep 08 2020 *)

A171913_vec(N, a=3, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 15 2019

from itertools import count, islice
def A171913gen(): # generator of terms
    b, bdict = 3, {3:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A171913_list = list(islice(A171913gen(),20)) # Chai Wah Wu, Dec 21 2021

a(n+1) = A181391(n) up to the first occurrence of a(1) = 3 in A181391. - M. F. Hasler, Jun 15 2019

Name edited and cross-references added by M. F. Hasler, Jun 15 2019

A171914 Van Eck sequence (cf. A181391) starting with a(1) = 4.

Original entry on oeis.org

4, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 17, 0, 6, 5, 5, 1, 14, 0, 6, 6, 1, 5, 7, 0, 6, 5, 4, 17, 17, 1, 9, 0, 8, 0, 2, 28, 0, 3, 0, 2, 5, 15, 0, 4, 17, 16, 0, 4, 4, 1, 20, 0, 5, 12, 0, 3, 18, 0, 3, 3, 1, 11, 0, 5, 11, 3, 6, 42, 0, 6, 3, 5, 8, 40, 0, 6, 6, 1, 17, 34, 0, 6

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty, i.e., a(n) does not appear earlier in the sequence. - M. F. Hasler, Jun 15 2019

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

A171914_vec(N, a=4, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 15 2019

from itertools import count, islice
def A171914gen(): # generator of terms
    b, bdict = 4, {4:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A171914_list = list(islice(A171914gen(),20)) # Chai Wah Wu, Dec 21 2021

a(n+1) = A181391(n) up to the first occurrence of a(1) = 4 in A181391. - M. F. Hasler, Jun 15 2019

Name edited and cross-references added by M. F. Hasler, Jun 15 2019

A171916 Van Eck's sequence (cf. A181391) starting with a(1) = 6.

Original entry on oeis.org

6, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 10, 0, 6, 3, 0, 3, 2, 9, 0, 4, 0, 2, 5, 0, 3, 9, 8, 0, 4, 9, 4, 2, 10, 22, 0, 7, 0, 2, 6, 26, 0, 4, 11, 0, 3, 20, 0, 3, 3, 1, 41, 0, 5, 30, 0, 3, 7, 21, 0, 4, 18, 0, 3, 7, 7, 1, 16, 0, 6, 30, 16, 4, 12, 0, 6, 6, 1, 11, 35, 0, 6, 5, 29, 0, 4, 13, 0, 3, 25, 0

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

Van Eck's sequence is defined by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty, i.e., a(n) does not appear earlier in the sequence. - M. F. Hasler, Jun 15 2019

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

A171916_vec(N, a=6, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 15 2019

from itertools import count, islice
def A171916gen(): # generator of terms
    b, bdict = 6, {6:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A171916_list = list(islice(A171916gen(),20)) # Chai Wah Wu, Dec 21 2021

Name edited and cross-references added by M. F. Hasler, Jun 15 2019

A171952 Positions of 2's in A181391.

Original entry on oeis.org

5, 7, 8, 14, 23, 40, 71, 79, 302, 460, 466, 468, 469, 714, 720, 722, 723, 934, 1018, 1157, 1456, 1641, 1660, 1675, 1708, 1727, 1844, 2157, 2356, 2470, 2583, 2589, 2591, 2592, 2744, 2829, 3116, 3381, 3388, 3390, 3391, 3568, 3623, 3655, 3696

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..835

Cf. A181391, A171865, A171951-A171956.

A171953 Positions of 3's in A181391.

Original entry on oeis.org

20, 22, 28, 33, 38, 46, 49, 50, 61, 68, 70, 76, 78, 86, 94, 103, 201, 204, 205, 217, 230, 256, 259, 260, 281, 327, 369, 391, 394, 395, 403, 422, 492, 550, 575, 583, 672, 675, 676, 740, 804, 815, 843, 1103, 1165, 1488, 1491, 1492, 1543, 1668

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..143

Cf. A181391, A171865, A171951-A171956.

A171954 Positions of 4's in A181391.

Original entry on oeis.org

17, 26, 58, 65, 107, 120, 127, 131, 132, 140, 144, 145, 154, 158, 159, 166, 185, 189, 190, 214, 220, 234, 238, 239, 248, 267, 290, 299, 301, 323, 326, 336, 340, 341, 350, 359, 373, 382, 434, 438, 439, 446, 452, 458, 464, 477, 481, 482, 496

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1035

Cf. A181391, A171865, A171951-A171956.

A171955 Positions of 5's in A181391.

Original entry on oeis.org

12, 16, 19, 34, 37, 39, 54, 91, 112, 123, 136, 175, 253, 286, 295, 318, 322, 355, 378, 386, 408, 413, 414, 526, 543, 555, 572, 620, 669, 737, 759, 768, 818, 865, 874, 895, 900, 901, 926, 942, 947, 948, 956, 1043, 1086, 1148, 1170, 1175, 1176

Offset: 1

N. J. A. Sloane, Oct 29 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..4802

Cf. A181391, A171865, A171951-A171956.

cp's OEIS Frontend

A358258 First n-bit number to appear in Van Eck's sequence (A181391).

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A358259 Positions of the first n-bit number to appear in Van Eck's sequence (A181391).

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A171912 Van Eck sequence (cf. A181391) starting with 2.

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A171913 Van Eck sequence (cf. A181391) starting with a(1) = 3.

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A171914 Van Eck sequence (cf. A181391) starting with a(1) = 4.

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A171916 Van Eck's sequence (cf. A181391) starting with a(1) = 6.

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A171952 Positions of 2's in A181391.

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A171953 Positions of 3's in A181391.

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A171954 Positions of 4's in A181391.