A357082 -id:A357082 - cp's OEIS frontend

A357148 a(n) = A357082(n-1) + A357082(n).

1, 3, 5, 7, 9, 15, 16, 15, 16, 24, 29, 32, 33, 29, 34, 29, 32, 36, 34, 42, 61, 64, 34, 32, 61, 64, 61, 64, 61, 64, 65, 72, 76, 64, 72, 85, 76, 64, 72, 82, 64, 72, 100, 104, 100, 91, 64, 72, 64, 72, 104, 100, 116, 127, 128, 129, 133, 128, 129, 128, 129, 128, 129

Offset: 1

Michael De Vlieger, Sep 15 2022

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Bitmap of (a(n-1) + a(n)), n = 1..2^11, 12X vertical exaggeration, read horizontally where black represents 1 and white 0, with least significant bit on bottom.
Michael De Vlieger, Let sequence b list primitive terms in this sequence. Annotated plot a(n) = b(k) at (n, k) for n = 1..84.
Michael De Vlieger, Plot a(n) = b(k) at (n, k) for n = 1..2^12.
Scott R. Shannon, Image for n = 0..2250000.

Cf. A357082.

nn = 120; c[] = False; j = a[0] = 0; u = 1; w = "0"; Do[k = u; While[Or[c[k], StringContainsQ[w, Set[v, IntegerString[j + k, 2]]]], k++]; Set[{a[n], c[k], b[n]}, {k, True, j + k}]; Set[{j, w}, {k, w <> IntegerString[k, 2]}]; If[k == u, While[c[u], u++]], {n, nn}]; Array[b, nn] (* _Michael De Vlieger, Sep 15 2022 *)

from itertools import islice
def agen():
    aset, astr, an, mink = {0}, "0", 0, 1
    while True:
        k = mink
        while k in aset or bin(an+k)[2:] in astr: k += 1
        while mink in aset: mink += 1
        yield an+k; an = k; aset.add(an); astr += bin(an)[2:]
print(list(islice(agen(), 63))) # Michael S. Branicky, Sep 16 2022

A357149 a(n) = smallest missing number in A357082(k) for k = 0..n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45

Offset: 0

Michael De Vlieger, Sep 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Plot of a(n), n = 1..2^10 in gold, with A357082(n) = b(n) in black, records in b(n) in red, local minima in b(n) in blue.
Rémy Sigrist, PARI program

Cf. A357082.

nn = 2^10; c[] = False; j = a[0] = 0; u = 1; w = "0"; Do[k = u; While[Or[c[k], StringContainsQ[w, Set[v, IntegerString[j + k, 2]]]], k++]; Set[{a[n], c[k], b[n]}, {k, True, u}]; Set[{j, w}, {k, w <> IntegerString[k, 2]}]; If[k == u, While[c[u], u++]], {n, nn}]; Array[b, nn] (* _Michael De Vlieger, Sep 15 2022 *)

PARI
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See Links section.
```

A357166 If n appears in A357082, then a(n) is the unique k such that A357082(k) = n; otherwise a(n) = -1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 23, 8, 6, 16, 11, 13, 49, 18, 14, 10, 15, 19, 12, 17, 47, 20, 24, 41, 22, 26, 34, 38, 28, 29, 31, 30, 27, 37, 33, 25, 21, 40, 32, 36, 46, 39, 35, 82, 51, 42, 78, 45, 48, 44, 74, 43, 52, 65, 67, 69, 50, 62, 60, 58, 53, 55, 87, 54, 56, 57

Offset: 0

Rémy Sigrist, Sep 16 2022

			A357082(42) = 47, hence a(47) = 42.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Rémy Sigrist, Perl program

Cf. A357082.

PARI
```
See Links section.
```
Perl
```
See Links section.
```

from itertools import islice
def agen():
    aset, appearsat, astr, an, mink, nn = {0}, {0: 0}, "0", 0, 1, 0
    for n in count(1):
        k = mink
        while k in aset or bin(an+k)[2:] in astr: k += 1
        while mink in aset: mink += 1
        an = k; aset.add(an); astr += bin(an)[2:]; appearsat[an] = n
        while nn in appearsat: yield appearsat[nn]; nn += 1
print(list(islice(agen(), 68))) # Michael S. Branicky, Sep 16 2022

A355611 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of |a(n) - a(n-1)| does not appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 3, 5, 9, 17, 7, 23, 2, 12, 22, 6, 16, 37, 58, 10, 38, 4, 32, 60, 14, 48, 82, 8, 42, 85, 15, 61, 107, 11, 67, 131, 18, 86, 13, 77, 141, 21, 89, 25, 93, 20, 84, 148, 19, 83, 147, 27, 91, 155, 26, 90, 154, 24, 88, 152, 28, 92, 156, 36, 100, 164, 30, 94, 158, 29, 142, 78, 191, 31, 95, 159

Offset: 0

Scott R. Shannon, Sep 12 2022

The sequence is conjectured to be a permutation of the positive integers. In the first 200000 terms the only fixed points are 1199 and 14767. It is unknown if more exist.

			a(5) = 17 as the concatenation of a(0)..a(4) in binary is "01111011001" and |17 - a(4)| = |17 - 9| = 8 = 1000_2 which does not appear in the concatenated string. Since 1 = 1_2, 2 = 10_2, 3 = 11_2, 4 = 100_2, 5 = 101_2, 6 = 110_2, 7 = 111_2 all appear in the concatenated string, a(5) cannot be less than 17.

Scott R. Shannon, Image of n=0..200000. The green line is a(n) = n.

Cf. A357377 (base 10), A357082, A007088, A030302, A118248, A341766.

from itertools import count, islice
def agen(): # generator of terms
    alst, aset, astr, an, mink, mindiff = [], set(), "", 0, 1, 1
    for n in count(0):
        yield an; aset.add(an); astr += bin(an)[2:]
        prevan, an = an, mink
        while an + mindiff <= prevan and (an in aset or bin(abs(an-prevan))[2:] in astr): an += 1
        if an in aset or bin(abs(an-prevan))[2:] in astr:
            an = max(mink, prevan + mindiff)
            while an in aset or bin(an-prevan)[2:] in astr:
                an += 1
        while mink in aset: mink += 1
        while bin(mindiff)[2:] in astr: mindiff += 1
print(list(islice(agen(), 72))) # Michael S. Branicky, Oct 05 2022

A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

Original entry on oeis.org

1, 10, 98, 767, 111, 122, 2, 11, 100, 889, 110, 4490, 400, 560, 1096, 124, 20, 129, 70, 502, 93, 171, 212, 361, 512, 26, 21, 36, 54, 14, 1011, 139, 99, 59, 550, 684, 345, 102, 1021, 1999, 2871, 137, 892, 89, 126, 875, 510, 994, 586, 2012, 662, 1836, 201, 405, 388, 2007, 2798, 1641, 50, 340

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 07 2023

In the first 10000 terms the smallest number that has not yet appeared is 696; it is therefore likely all numbers eventually appear although this is unknown.

			a(2) = 10 as a(1) + 10 = 1 + 10 = 11 which is a substring of "1" + "10" = "110".
a(3) = 98 as a(1) + a(2) + 98 = 1 + 10 + 98 = 109 which is a substring of "1" + "10" + "98" = "11098".
a(4) = 767 as a(1) + a(2) + a(3) + 767 = 1 + 10 + 98 + 767 = 876 which is a substring of "1" + "10" + "98" + "767" = "11098767".

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Eric Angelini, Échecs et Maths, Personal blog, July 2023.

Cf. A359482, A300000, A339144, A357082.

from itertools import islice
def agen(): # generator of terms
    s, mink, aset, concat = 1, 2, {1}, "1"
    yield from [1]
    while True:
        an = mink
        while an in aset or not str(s+an) in concat+str(an): an += 1
        aset.add(an); s += an; concat += str(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Feb 08 2024

A357377 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that |a(n) - a(n-1)| does not appear in the string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 2, 6, 10, 14, 22, 4, 12, 20, 28, 44, 8, 24, 40, 56, 18, 34, 50, 23, 39, 55, 26, 42, 58, 29, 45, 61, 31, 47, 63, 27, 43, 59, 75, 37, 53, 69, 85, 36, 52, 68, 30, 46, 62, 78, 94, 110, 33, 49, 65, 81, 97, 113, 41, 57, 73, 35, 51, 67, 105, 143, 16, 54

Offset: 0

Scott R. Shannon, Sep 26 2022

The sequence is conjectured to be a permutation of the positive integers. In the first 200000 terms the only fixed points are 20, 24, 43 and 115. It is likely no more exist although this is unknown.

There are no other fixed points in the first 870000 terms. - Michael S. Branicky, Oct 05 2022

			a(12) = 25 as the concatenation of a(0)..a(11) is "013579111315171921" and |25 - a(11)| = |25 - 21| = 4 which does not appear in the concatenated string. Since a(11) contains a '2' and all other odd numbers appear in the string a(12) cannot be 23 or any even number less than 25.
a(13) = 2 as the concatenation of a(0)..a(12) is "01357911131517192125" and |2 - a(12)| = |2 - 25| = 23 which does not appear in the concatenated string.

Scott R. Shannon, Image for n=0..200000. The green line is a(n) = n.

Cf. A355611 (base 2), A357082, A007088, A030302, A118248, A341766.

from itertools import count, islice
def agen(): # generator of terms
    alst, aset, astr, an, mink, mindiff = [], set(), "", 0, 1, 1
    for n in count(0):
        yield an; aset.add(an); astr += str(an)
        prevan, an = an, mink
        while an + mindiff <= prevan and (an in aset or str(abs(an-prevan)) in astr): an += 1
        if an in aset or str(abs(an-prevan)) in astr:
            an = max(mink, prevan + mindiff)
            while an in aset or str(an-prevan) in astr:
                an += 1
        while mink in aset: mink += 1
        while str(mindiff) in astr: mindiff += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, Oct 05 2022

A357449 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of a(n) plus the largest previous term does not appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 7, 9, 14, 15, 16, 17, 18, 20, 12, 24, 8, 28, 26, 30, 22, 33, 11, 21, 31, 32, 36, 37, 27, 35, 41, 13, 23, 40, 44, 38, 62, 46, 66, 19, 42, 63, 65, 69, 39, 59, 60, 68, 72, 56, 57, 71, 76, 52, 53, 80, 48, 49, 55, 58, 61, 64, 83, 45, 73, 77, 81, 82, 85, 43, 50, 75, 79, 87, 51

Offset: 0

Scott R. Shannon, Sep 29 2022

The main concentration of terms lies near the line a(n) = n; there are 26 fixed points in the first 100000 terms. The sequence is conjectured to be a permutation of the positive integers.

			a(9) = 9 as the concatenation of a(0)..a(8) in binary is "0110111001011010110111" and 9 plus the largest previous term = 9 + 10 = 19 = 10011_2 which does not appear in the concatenated string. Since 10 + 8 = 18 = 10010_2 appears in the concatenated string, a(9) cannot be 8.

Scott R. Shannon, Image for n = 0..100000.

Cf. A357082, A355611, A007088, A030302, A118248, A341766.

from itertools import islice
def agen():
    aset, astr, an, mink = {0}, "0", 0, 1
    while True:
        yield an; k, m = mink, max(aset)
        while k in aset or bin(m+k)[2:] in astr: k += 1
        while mink in aset: mink += 1
        an = k; aset.add(an); astr += bin(an)[2:]
print(list(islice(agen(), 77))) # Michael S. Branicky, Sep 29 2022

A357150 Primitive terms in A357148.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 16, 24, 29, 32, 33, 34, 36, 42, 61, 64, 65, 72, 76, 82, 85, 91, 100, 104, 116, 127, 128, 129, 133, 144, 146, 153, 154, 169, 172, 179, 192, 209, 224, 256, 257, 258, 260, 262, 264, 270, 276, 281, 303, 322, 325, 339, 355, 356, 360, 400, 417, 418

Offset: 1

Michael De Vlieger, Sep 15 2022

nonn

			Let b(n) = A357082(n).
3 is in the sequence since S = b(1) + b(2) = 1 + 2 = 3. Since b(3) = 3, it is not possible to see S = 3 again.
4 is not in the sequence since no sum S = 4 appears before b(4) = 4 = "100" in binary, whereafter "100" is appended to W, and thereafter prohibited as a sum of adjacent terms in b for n > 4.
32 is in the sequence since S = b(11) + b(12) = b(16) + b(17) = b(23) + b(24) = 32. We note that b(31) = 32, therefore these are the only instances of sum S = 32.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Bitmap of a(n), n = 1..2^10, 6X vertical exaggeration, read horizontally where black represents 1 and white 0, with least significant bit on bottom.
Scott R. Shannon, Image for n=1..28000.

Cf. A357082, A357148.

nn = 650; s[] = c[] = False; j = 0; i = u = 1; w = "0"; b = Reap[Do[k = u; While[Or[c[k], StringContainsQ[w, Set[v, IntegerString[j + k, 2]]]], k++]; c[k] = True; Sow[k]; If[! s[#], Set[{a[i], s[#]}, {#, True}]; i++] &[j + k]; Set[{j, w}, {k, w <> IntegerString[k, 2]}]; If[k == u, While[c[u], u++]], {n, nn}] ][[-1, -1]]; TakeWhile[Array[a, i - 1], MemberQ[b, #] &]

A359858 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the ones' complement of the binary string of a(n-1) + a(n) does not appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 2, 6, 5, 3, 8, 14, 18, 20, 12, 24, 23, 9, 27, 37, 10, 22, 25, 7, 40, 39, 52, 42, 49, 15, 32, 59, 35, 56, 38, 53, 41, 50, 44, 47, 81, 13, 78, 16, 75, 57, 34, 94, 61, 30, 98, 60, 31, 97, 58, 33, 122, 63, 28, 127, 55, 36, 126, 65, 118, 67, 95, 88, 74, 109, 76, 86, 99, 84, 101, 82, 80, 103, 187

Offset: 0

Scott R. Shannon, Jan 16 2023

In the first 100000 terms the fixed points are 55, 123, 1779, 2009, although it is likely more exist. The sequence is conjectured to be a permutation of the positive integers. Note that 1 does not appear until the 160th term.

			a(3) = 5 as the concatenation of a(0)..a(2) in binary is "010110" and a(2) + 5 = 6 + 5 = 11 = 1011_2 whose ones' complement = 100_2, which does not appear in the concatenated string. Note the ones' complements of 6+1, 6+3, 6+4 are 0_2, 110_2, 101_2, all of which appear in the concatenated string.

Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n. Note some of the patterns appearing are reminiscent of those in the images of A357082.

Cf. A035327, A007088, A357082, A355611.

A360521 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that neither the binary string a(n-1) + a(n) nor the same string reversed appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 9, 7, 17, 15, 18, 14, 19, 13, 20, 12, 21, 11, 22, 26, 8, 24, 27, 37, 28, 23, 25, 39, 29, 35, 30, 34, 31, 33, 32, 36, 41, 44, 43, 42, 38, 62, 57, 70, 58, 69, 59, 68, 60, 67, 61, 66, 63, 64, 65, 71, 76, 52, 84, 72, 56, 80, 48, 88, 40, 96, 51, 77, 83, 45, 91, 81, 47, 89, 86

Offset: 0

Scott R. Shannon, Feb 09 2023

As in A357082 the main concentration of terms is along the line a(n) = n, so there are numerous fixed points - there are 24 fixed points in the first 200000 terms. The sequence is conjectured to be a permutation of the positive integers.

Note that when the binary string of a(n-1) + a(n) is reversed any resulting leading 0's are retained for the string comparison.

			a(11) = 15 as the concatenation of a(0)..a(10) in binary is "0110111001011010110100111110001" and a(10) + 15 = 17 + 15 = 32 = 100000_2 which does not appear in the concatenated string, nor does its reverse "000001". Although 17 + 12 = 29 = 11101_2 does not appear in the string its reverse "10111" does, so a(11) cannot be 12. This is the first term to differ from A357082.

Michael S. Branicky, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image for n=0..60000.
Scott R. Shannon, Image for n=0..200000. The tip of the large right-pointing 'arrow' corresponds to the term a(159580) = 131072 = 100000000000000000_2.
Scott R. Shannon, Image for n=0..600000.
Scott R. Shannon, Image of first million terms.
Scott R. Shannon and Zach J. Shannon, Image of first ten million terms. This is a high resolution image so zoom in to see the details.
Scott R. Shannon and Zach J. Shannon, Image of first thirty six million terms. This is a high resolution image so zoom in to see the details.

Cf. A357082, A007088, A030302, A118248, A341766.

nn = 76; c[] = False; j = a[0] = 0; u = 1; w = "0"; Do[k = u; While[Or[c[k], StringContainsQ[w, Set[v, IntegerString[j + k, 2]]], StringContainsQ[w, StringReverse[v]]], k++]; Set[{a[n], c[k]}, {k, True}]; Set[{j, w}, {k, w <> IntegerString[k, 2]}]; If[k == u, While[c[u], u++]], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger, Feb 15 2023 *)

from itertools import islice
def agen(): # generator of terms
    an, astr, aset, mink = 0, "", set(), 1
    while True:
        yield an
        s = bin(an)[2:]
        astr += s
        k = mink
        while k in aset or (t:=bin(an+k)[2:]) in astr or t[::-1] in astr:
            k += 1
        an = k
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, Feb 10 2023

cp's OEIS Frontend

A357148 a(n) = A357082(n-1) + A357082(n).

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A357149 a(n) = smallest missing number in A357082(k) for k = 0..n.

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PARI

A357166 If n appears in A357082, then a(n) is the unique k such that A357082(k) = n; otherwise a(n) = -1.

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A355611 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of |a(n) - a(n-1)| does not appear in the binary string concatenation of a(0)..a(n-1).

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A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

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A357377 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that |a(n) - a(n-1)| does not appear in the string concatenation of a(0)..a(n-1).

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A357449 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of a(n) plus the largest previous term does not appear in the binary string concatenation of a(0)..a(n-1).

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A357150 Primitive terms in A357148.

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A359858 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the ones' complement of the binary string of a(n-1) + a(n) does not appear in the binary string concatenation of a(0)..a(n-1).