A373345 -id:A373345 - cp's OEIS frontend

A373346 Row sums of A373345.

1, 2, 3, 5, 4, 6, 7, 5, 7, 8, 9, 12, 6, 8, 9, 10, 13, 11, 14, 15, 7, 9, 10, 11, 14, 12, 15, 16, 13, 16, 17, 18, 22, 8, 10, 11, 12, 15, 13, 16, 17, 14, 17, 18, 19, 23, 15, 18, 19, 20, 24, 21, 25, 26, 9, 11, 12, 13, 16, 14, 17, 18, 15, 18, 19, 20, 24, 16, 19, 20, 21, 25

Offset: 1

Paolo Xausa, Jun 01 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A373345.

Join[{1}, Map[Total[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 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

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).

Original entry on oeis.org

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 &]]]

A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 09 2024

A strong Schreier set is a subset of the positive integers with cardinality less than the minimum element in the set (see Chu link).
Each term k of 2*A371176 can be put into a one-to-one correspondence with a strong 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 strong Schreier set.
Arranging the elements in each set in decreasing order results in the sets being listed in lexicographical order (see example). Cf. A373579 for the elements arranged in increasing order.
The number of sets having maximum element m is A000045(m-1).

			Triangle begins:
                                        Corresponding
   n  A371176(2*n)  bin(A371176(2*n))   strong Schreier set
                                        (this sequence)
  ---------------------------------------------------------
   1        2               10          {2}
   2        4              100          {3}
   3        8             1000          {4}
   4       12             1100          {4, 3}
   5       16            10000          {5}       Sets are
   6       20            10100          {5, 3}    lexicographically
   7       24            11000          {5, 4}    ordered
   8       32           100000          {6}
   9       36           100100          {6, 3}
  10       40           101000          {6, 4}
  11       48           110000          {6, 5}
  12       56           111000          {6, 5, 4}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
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, A007895 (conjectured row lengths), A371176, A373556, A373579, A373853 (row sums).

Join[{{2}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 400, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373345(n,k) + 1.

A373347 Positive integers k such that A000120(k) > A001511(k).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Paolo Xausa, Jun 01 2024

Numbers whose binary expansion does not encode for any Schreier set (cf. A371176 and A373345).

All odd numbers > 1 are terms.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Complement of A371176.

Cf. A000120, A001511, A008466, A373345, A373360 (first differences).

Select[Range[100], DigitSum[#, 2] > IntegerExponent[#, 2] + 1 &]

isok(k) = hammingweight(k) > valuation(2*k, 2); \\ Michel Marcus, Jun 07 2024

def isa(n): return (n - 1).bit_count() < ((n.bit_count() - 1) << 1)
print([n for n in range(100) if isa(n)])  # Peter Luschny, Jun 07 2024

a(k) = 2^(n+1) - 1; a(k+1) = 2^(n+1) + 1, where k = A008466(n+1).

A373359 Irregular triangle read by rows where row n lists (in increasing order) the elements of the Schreier set encoded by A371176(n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 04 2024

See A373345 (where elements in each set are listed in decreasing order) for more information.

			Triangle begins:
                                   Corresponding Schreier
   n  A371176(n)  bin(A371176(n))  set (this sequence)
  -------------------------------------------------------
   1      1              1         {1}
   2      2             10         {2}
   3      4            100         {3}
   4      6            110         {2, 3}
   5      8           1000         {4}
   6     10           1010         {2, 4}
   7     12           1100         {3, 4}
   8     16          10000         {5}
   9     18          10010         {2, 5}
  10     20          10100         {3, 5}
  11     24          11000         {4, 5}
  12     28          11100         {3, 4, 5}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.

Cf. A371176, A373345, A373558, A373579.

Cf. A007895 (conjectured row lengths), A373346 (row sums), A373347.

Join[{{1}}, Map[PositionIndex[Reverse[IntegerDigits[#, 2]]][1] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373579(n,k) - 1.

A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

Original entry on oeis.org

2, 1, 3, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 7, 9, 1, 2, 3, 6, 11, 14, 1, 2, 3, 6, 10, 17, 22, 1, 2, 3, 5, 10, 17, 26, 35, 1, 2, 3, 5, 10, 16, 28, 40, 56, 1, 2, 3, 5, 8, 16, 26, 45, 62, 90, 1, 2, 3, 5, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234

Offset: 1

Paolo Xausa, Jun 21 2024

			The array begins:
  k\n|  1  2  3  4   5   6   7   8   9   10  ...
  ----------------------------------------------
   1 |  2, 3, 4, 6,  9, 14, 22, 35, 56,  90, ... = A001611 (from n = 2).
   2 |  1, 2, 4, 7, 11, 17, 26, 40, 62,  97, ...
   3 |  1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
   4 |  1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
   5 |  1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
   6 |  1, 2, 3, 5,  8, 16, 26, 42, 68, 110, ...
   7 |  1, 2, 3, 5,  8, 13, 26, 42, 68, 110, ...
   8 |  1, 2, 3, 5,  8, 13, 21, 42, 68, 110, ...
   9 |  1, 2, 3, 5,  8, 13, 21, 34, 68, 110, ...
  10 |  1, 2, 3, 5,  8, 13, 21, 34, 55, 110, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
Hung Viet Chu and Zachary Louis Vasseur, Weighted Schreier-type Sets and the Fibonacci Sequence, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.

Cf. A000045, A001611, A373345, A373556, A373557.

A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]

T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;
T(k,n) = A000045(n+1) otherwise.
T(n,n) = 2*A000045(n).

cp's OEIS Frontend

A373346 Row sums of A373345.

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

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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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A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

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A373347 Positive integers k such that A000120(k) > A001511(k).

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A373359 Irregular triangle read by rows where row n lists (in increasing order) the elements of the Schreier set encoded by A371176(n).

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A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

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