A118248 -id:A118248 - cp's OEIS frontend

A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

0, 0, 0, 1, 0, 1, 2, 2, 0, 1, 0, 3, 2, 3, 4, 3, 0, 1, 1, 2, 1, 2, 0, 6, 2, 3, 3, 5, 5, 4, 6, 4, 0, 1, 1, 2, 0, 3, 2, 3, 1, 3, 1, 4, 1, 3, 3, 8, 2, 3, 4, 4, 3, 5, 3, 8, 5, 5, 5, 6, 8, 5, 8, 5, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 1, 5, 3, 4, 1, 3, 2, 5, 2, 4, 2, 6, 1, 4, 3, 6, 2, 6, 4, 10, 2, 3, 4, 4, 3

Offset: 0

M. F. Hasler, Jan 03 2013

nonn

Related to "early bird" (decimal: A116700, binary: A161373) and Hannah Rollman's numbers (cf. A048991, A048992 for decimal; A118248 and A118247-A118251 for binary versions). The latter would correspond to a variant of this sequence which has indices of nonzero terms omitted from the concatenation.

			a(3) = 1 since concatenation of 0,1,2 in binary yields "0110", and 3 = "11"[2] occurs once in this string.

(nMax)->my(c=[],cnt(t,s,M)=M=2^#s-1;sum(i=0,#t-#s,vecextract(t,M<

A190784 Numbers whose binary representation is a substring of the concatenation of the binary representation of all smaller nonnegative integers not listed earlier, taken in decreasing order.

Original entry on oeis.org

2, 6, 7, 9, 11, 12, 14, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 33, 34, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 72, 73, 74, 77, 78, 80, 81, 82, 84, 86, 87, 89, 90, 92, 93, 94

Offset: 1

M. F. Hasler, Dec 29 2012

Also, nonnegative integers which do not occur in A118250.

Up to the reversed (decreasing) order of concatenation, a binary analog of Hannah Rollman's numbers A048992.

			The binary representation of 2="10"[2] is a substring of the concatenation of 1 and 0, therefore a(1)=2. This term a(1)=2="10" will henceforth be excluded from the concatenations considered in the sequel.
The binary representations of 3, 4 and 5 are not a substrings of concat("1", "0") resp. concat("11", "1", "0") resp. concat("100", "11", "1", "0"). (Note that 2="10" is not among the concatenated numbers.)
But 6="110"[2] is again a substring of concat(5="101", 4="100", 3="11", "1", "0"), therefore a(2)=6. In the sequel, a(2)=6="110" will now also be always excluded from the concatenations, as is a(1)=2.

Analog of A128291 for the "with reversal" variant A118250 of A118248.

Cf. A048991, A048992.

A190897 Concatenation of A190896 written in binary.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0

Offset: 0

M. F. Hasler, Dec 29 2012

This sequence is mainly included because the sequences A118248, A118250, A118252 (variants of A190896) had "historically" been defined through the respective analogs A118247, A118249, A118251 of this present ("binary") sequence.

Cf. A118247, A118249, A118251 (variants with nonnegative integers and/or binary representations reversed).

A309340 a(n) is the least nonnegative number whose binary digits do not appear in order (not necessarily consecutively) in the concatenation of the binary representations of all previous terms.

Original entry on oeis.org

0, 1, 2, 4, 9, 38, 304, 2433, 38938, 2492040, 318981139, 163318343502

Offset: 1

Rémy Sigrist, Jul 24 2019

This sequence is a variant of A118248.

a(13) > 4.5*10^12. - Giovanni Resta, Aug 08 2019

See A308540 for the decimal variant.

Cf. A118248.

a(12) from Giovanni Resta, Aug 08 2019

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

A357482 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of the number of 1's in the binary value of a(n) + the number of 1's in the binary values of all previous terms does not appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 2, 3, 7, 4, 5, 63, 8, 6, 9, 16, 127, 11, 10, 12, 13, 14, 19, 511, 1023, 15, 21, 17, 31, 18, 20, 22, 24, 25, 33, 23, 27, 26, 28, 35, 37, 38, 41, 1535, 29, 30, 32, 34, 47, 36, 40, 55, 39, 43, 42, 45, 255, 46, 51, 383, 48, 44, 4095, 64, 447, 65, 95, 53, 191, 767, 1791, 59, 49, 54, 57, 50, 52

Offset: 0

Scott R. Shannon, Sep 30 2022

The sequence contains large jumps in value due to some terms having to be 1 less than a power of 2 to contain sufficient 1's in their binary value to meet the term selection criteria. For example a(386) = 512, a(387) = 68719476735. See the examples below.

			a(7) = 63 as 63 = 111111_2 which contains six 1's, the concatenation of the binary values of a(0)..a(6) is "011011111100101" which contains ten 1's, and 6 + 10 = 16 = 10000_2 which does not appear in the concatenated binary string of previous terms. All smaller unused numbers less than 63 have one to five 1's in their binary values leading to sums of 11, 12, 13, 14 or 15, but the binary values of these five sums all appear in the concatenated binary string of previous terms.

Cf. A357082, A357449, A355611, A007088, A030302, A118248.

cp's OEIS Frontend

A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

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A190784 Numbers whose binary representation is a substring of the concatenation of the binary representation of all smaller nonnegative integers not listed earlier, taken in decreasing order.

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A190897 Concatenation of A190896 written in binary.

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A309340 a(n) is the least nonnegative number whose binary digits do not appear in order (not necessarily consecutively) in the concatenation of the binary representations of all previous terms.

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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).

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A357482 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of the number of 1's in the binary value of a(n) + the number of 1's in the binary values of all previous terms does not appear in the binary string concatenation of a(0)..a(n-1).

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