A355489 -id:A355489 - cp's OEIS frontend

A373556 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in decreasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

1, 3, 2, 4, 2, 5, 2, 5, 4, 3, 6, 2, 6, 4, 3, 6, 5, 3, 7, 2, 7, 4, 3, 7, 5, 3, 7, 6, 3, 7, 6, 5, 4, 8, 2, 8, 4, 3, 8, 5, 3, 8, 6, 3, 8, 6, 5, 4, 8, 7, 3, 8, 7, 5, 4, 8, 7, 6, 4, 9, 2, 9, 4, 3, 9, 5, 3, 9, 6, 3, 9, 6, 5, 4, 9, 7, 3, 9, 7, 5, 4, 9, 7, 6, 4, 9, 8, 3

Offset: 1

Paolo Xausa, Jun 09 2024

A maximal Schreier set is a subset of the positive integers with cardinality equal to the minimum element in the set (see Chu link).
For n >= 2, each term k = 2*A355489(n-1) can be put into a one-to-one correspondence with a maximal Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding maximal Schreier set.
See A373558 for the elements in each set arranged in increasing order.
The number of sets having maximum element m (for m >= 2) is A000045(m-2).

			Triangle begins:
                                           Corresponding
   n  2*A355489(n-1)  bin(2*A355489(n-1))  maximal Schreier set
                                           (this sequence)
  ---------------------------------------------------------------
   1                                       {1}
   2         6                 110         {3, 2}
   3        10                1010         {4, 2}
   4        18               10010         {5, 2}
   5        28               11100         {5, 4, 3}
   6        34              100010         {6, 2}
   7        44              101100         {6, 4, 3}
   8        52              110100         {6, 5, 3}
   9        66             1000010         {7, 2}
  10        76             1001100         {7, 4, 3}
  11        84             1010100         {7, 5, 3}
  12       100             1100100         {7, 6, 3}
  13       120             1111000         {7, 6, 5, 4}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10003 (rows 1..1892 of the triangle, flattend).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373345.

Cf. A000045, A143299 (conjectured row lengths), A355489, A373557, A373558, A373854 (row sums).

Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 500, 2], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 1 &]]]

A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 3, 4, 5, 2, 6, 3, 4, 6, 3, 5, 6, 2, 7, 3, 4, 7, 3, 5, 7, 3, 6, 7, 4, 5, 6, 7, 2, 8, 3, 4, 8, 3, 5, 8, 3, 6, 8, 4, 5, 6, 8, 3, 7, 8, 4, 5, 7, 8, 4, 6, 7, 8, 2, 9, 3, 4, 9, 3, 5, 9, 3, 6, 9, 4, 5, 6, 9, 3, 7, 9, 4, 5, 7, 9, 4, 6, 7, 9, 3, 8, 9

Offset: 1

Paolo Xausa, Jun 10 2024

See A373556 (where elements in each set are listed in decreasing order) for more information.

			Triangle begins:
                                           Corresponding
   n  2*A355489(n-1)  bin(2*A355489(n-1))  maximal Schreier set
                                           (this sequence)
  ---------------------------------------------------------------
   1                                       {1}
   2         6                 110         {2, 3}
   3        10                1010         {2, 4}
   4        18               10010         {2, 5}
   5        28               11100         {3, 4, 4}
   6        34              100010         {2, 6}
   7        44              101100         {3, 4, 6}
   8        52              110100         {3, 5, 6}
   9        66             1000010         {2, 7}
  10        76             1001100         {3, 4, 7}
  11        84             1010100         {3, 5, 7}
  12       100             1100100         {3, 6, 7}
  13       120             1111000         {4, 5, 6, 7}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10003 (rows 1..1892 of the triangle, flattend).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373359.

Cf. A143299 (conjectured row lengths), A355489, A373556, A373579, A373854 (row sums).

Join[{{1}}, Map[PositionIndex[Reverse[IntegerDigits[#, 2]]][1] &, Select[Range[2, 500, 2], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 1 &]]]

A371176 Numbers k such that A000120(k) <= A001511(k).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 32, 34, 36, 40, 44, 48, 52, 56, 64, 66, 68, 72, 76, 80, 84, 88, 96, 100, 104, 112, 120, 128, 130, 132, 136, 140, 144, 148, 152, 160, 164, 168, 176, 184, 192, 196, 200, 208, 216, 224, 232, 240, 256, 258, 260, 264, 268

Offset: 1

Mikhail Kurkov, Mar 14 2024

It appears that this sequence is obtained when ordering Schreier sets as explained in the Bird link. See decM(n) PARI code. - Michel Marcus, May 31 2024
That is correct since the binary representation of these numbers can be put into 1-to-1 correspondence with Schreier sets, which satisfy |X| <= min X, using the indicator function of X as the bits (starting from the right, LSB). The reason is that A000120 then computes |X| and A001511 computes min X. For example, the Schreier set X = {2, 5} can be mapped to 10010_2 = 18. - Michael S. Branicky, May 31 2024
From David A. Corneth, May 31 2024: (Start)
If k is in the sequence then so is 2*k.
a(A000045(k)) = 2^(k-2) for k >= 2. (End)
Apart from a(1), all terms are even. - Paolo Xausa, May 31 2024
Zeckendorf representation of n with rewrite 0 -> 0, {0, 1} -> 1 and k-1 zeros appended to the right side (where k is the number of ones in the given representation) and then interpreted as binary expansion is the same as a(n) (see the first formula). - Mikhail Kurkov, Oct 21 2024

Michel Marcus, Table of n, a(n) for n = 1..10945
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Vaclav Kotesovec, Plot of log(a(n))/log(n) for n = 2..2129243
Vaclav Kotesovec, Plot of a(n) / 2^(log(sqrt(5)*n)/log((1 + sqrt(5))/2) - 2) for n = 2..2129243

Cf. A000045, A000120, A001316, A001511, A003849, A005206, A007895, A048679, A066628, A072649, A213911.
Cf. A355489, A373345, A373347 (complement), A373557.
Cf. A104287.

filter:= proc(n) convert(convert(n,base,2),`+`) <= 1+padic:-ordp(n,2) end proc:
select(filter, [1,seq(i,i=2..1000,2)]); # Robert Israel, Oct 20 2024

Join[{1}, Select[Range[2, 1000, 2], DigitSum[#, 2] <= IntegerExponent[#, 2] + 1 &]] (* Paolo Xausa, Aug 12 2025 *)

isok(n) = hammingweight(n) <= (valuation(n, 2) + 1)

M(n) = my(list=List()); for (i=1, n, forsubset(i, s, my(bOk = if (#s && (vecmax(s) == n), #s <= vecmin(s), 0)); if (bOk, listput(list, vecsort(Vec(s),,4))););); Vec(list);
decM(nn) = my(v = vector(nn, k, M(k)), list=List()); for (i=1, #v, my(vi = v[i]); for (j=1, #vi, my(s = vecsort(vi[j]), slist=List(), m = vecmax(s)); forstep(k=m, 1, -1, listput(slist, sign(vecsearch(s, k)))); listput(list, fromdigits(Vec(slist), 2)););); vecsort(Vec(list)); \\ Michel Marcus, May 31 2024

def ok(n): return n.bit_count() <= (-n&n).bit_length()
print([k for k in range(1, 300) if ok(k)]) # Michael S. Branicky, May 31 2024

# Assuming the list starts with 0.
def a():
    n = na = nb = 1
    while True:
        yield not(nb < (na - 1) << 1)
        nb, na = na, n.bit_count()
        n += 1
aList = a(); print([n for n in range(77) if next(aList)]) # Peter Luschny, Jun 07 2024

a(n) = b(n)*A001316(b(n))/2 where b(n) = A048679(n).
a(n) = Sum_{i=0..n-1} 2^A213911(i).
a(n) = 2^(A072649(n) - 1) + [c(n) > 0]*2*a(c(n)) where c(n) = A066628(n).
a(n) = 2*a(A005206(n)) - A003849(n)*2^A007895(n-1) for n > 1 with a(1) = 1.
Conjecture: lim sup_{n -> oo} log(a(n))/log(n) = log(2) / log((1 + sqrt(5))/2) = 1.440420090412556479... = A104287. - Vaclav Kotesovec, Aug 12 2025

A358654 a(n) = A025480(A353654(n+1) - 1).

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 15, 4, 11, 13, 14, 31, 9, 10, 23, 12, 27, 29, 30, 63, 8, 19, 21, 22, 47, 25, 26, 55, 28, 59, 61, 62, 127, 17, 18, 39, 20, 43, 45, 46, 95, 24, 51, 53, 54, 111, 57, 58, 119, 60, 123, 125, 126, 255, 16, 35, 37, 38, 79, 41, 42, 87, 44, 91, 93

Offset: 0

Mikhail Kurkov, Nov 25 2022

Permutation of the nonnegative integers.

Conjecture: A247648(n) with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n).

Cf. A025480, A048679, A247648, A343152, A348366, A353654, A355489.

Conjecture: a(n) = A348366(A343152(n)) for n > 0 with a(0) = 1.

cp's OEIS Frontend

A373556 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in decreasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

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A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

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A371176 Numbers k such that A000120(k) <= A001511(k).

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A358654 a(n) = A025480(A353654(n+1) - 1).

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