A371176 -id:A371176 - cp's OEIS frontend

A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

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

Offset: 1

Paolo Xausa, Jun 01 2024

A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.
Each term k of A371176 can be put into a one-to-one correspondence with a 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 Schreier set (see A371176 and Bird link).
See A373359 for the elements in each set arranged in increasing order.
The number of sets having maximum element m is A000045(m).

			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         {3, 2}
   5      8           1000         {4}
   6     10           1010         {4, 2}
   7     12           1100         {4, 3}
   8     16          10000         {5}
   9     18          10010         {5, 2}
  10     20          10100         {5, 3}
  11     24          11000         {5, 4}
  12     28          11100         {5, 4, 3}
  ...

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. A000045, A371176, A373359, A373556, A373557.

Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.

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

T(n,k) = A373557(n,k) - 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.

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.

A373579 Irregular triangle read by rows where row n lists (in increasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 10 2024

See A373557 (where elements in each set are listed in decreasing order) for more information.

			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          {3, 4}
   5       16            10000          {5}
   6       20            10100          {3, 5}
   7       24            11000          {4, 5}
   8       32           100000          {6}
   9       36           100100          {3, 6}
  10       40           101000          {4, 6}
  11       48           110000          {5, 6}
  12       56           111000          {4, 5, 6}
  ...

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 A373359.

Cf. A007895 (conjectured row lengths), A371176, A373557, A373558, A373853 (row sums).

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

T(n,k) = A373359(n,k) + 1.

A373303 First differences of A371176.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 8, 2, 2, 4, 4, 4, 4, 4, 8, 4, 4, 8, 8, 8, 2, 2, 4, 4, 4, 4, 4, 8, 4, 4, 8, 8, 8, 4, 4, 8, 8, 8, 8, 8, 16, 2, 2, 4, 4, 4, 4, 4, 8, 4, 4, 8, 8, 8, 4, 4, 8, 8, 8, 8, 8, 16, 4, 4, 8, 8, 8, 8, 8, 16, 8, 8, 16, 16

Offset: 1

Michel Marcus, May 31 2024

nonn

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

Cf. A371176.

Differences[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);
my(v=select(isok, [1..500])); vector(#v-1, k, v[k+1]-v[k])

a(n) = A371176(n+1) - A371176(n).

A104287 Decimal expansion of log base phi of 2.

Original entry on oeis.org

1, 4, 4, 0, 4, 2, 0, 0, 9, 0, 4, 1, 2, 5, 5, 6, 4, 7, 9, 0, 1, 7, 5, 5, 1, 4, 9, 9, 5, 8, 7, 8, 6, 3, 8, 0, 2, 4, 5, 8, 6, 0, 4, 1, 4, 2, 6, 8, 4, 0, 5, 6, 0, 8, 1, 6, 4, 5, 4, 4, 1, 7, 2, 9, 5, 6, 6, 5, 1, 3, 2, 8, 4, 3, 5, 2, 9, 9, 0, 3, 6, 7, 2, 7, 9, 5, 2, 8, 2, 2, 0, 4, 9, 7, 3, 5, 7, 5, 9, 1, 6, 3, 1, 2, 7

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 28 2005

The fractal dimension of the goldpoint snowflake (Turner, 2003). - Amiram Eldar, Jan 11 2022

			1.4404200904125564790175514995878638024586041426840560816454417295665...

Krassimir Atanassova, Vassia Atanassova, Anthony Shannon and John Turner, New Visual Perspectives on Fibonacci Numbers, World Scientific, 2002, p. 218.

Greg Kuperberg, Breaking the cubic barrier in the Solovay-Kitaev algorithm, QIP2023 video (2023).
J. C. Turner, Some fractals in goldpoint geometry, The Fibonacci Quarterly, Vol. 41, No. 1 (2003), pp. 63-71.
Index entries for transcendental numbers.

Cf. A001622, A002162, A002390, A371176.

RealDigits[Log[2]/Log[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Nov 24 2020 *)

log(2)/log((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 15 2019

Equals log(2) / log((sqrt(5)+1)/2).

Equals A002162/A002390. - Amiram Eldar, Nov 24 2020

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).

A355489 Numbers k such that A000120(k) = A007814(k) + 2.

Original entry on oeis.org

3, 5, 9, 14, 17, 22, 26, 33, 38, 42, 50, 60, 65, 70, 74, 82, 92, 98, 108, 116, 129, 134, 138, 146, 156, 162, 172, 180, 194, 204, 212, 228, 248, 257, 262, 266, 274, 284, 290, 300, 308, 322, 332, 340, 356, 376, 386, 396, 404, 420, 440, 452, 472, 488, 513, 518

Offset: 1

Mikhail Kurkov, Jul 04 2022 [verification needed]

Each term k, doubled, can be put into a one-to-one correspondence with a maximal Schreier set (a subset of the positive integers with cardinality equal to the minimum element in the set) by interpreting the 1-based position of the ones in the binary expansion of 2*k (where position 1 corresponds to the least significant bit) as the elements of the corresponding maximal Schreier set. See A373556 for more information. Cf. also A371176. - Paolo Xausa, Jun 13 2024

Cf. A000045, A000120, A007814, A010056, A025480, A048679, A072649, A371176, A373556.

Select[Range[500], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 2 &] (* Amiram Eldar, Jul 04 2022 *)

r=quadgen(5);
A355489_upto(nMax)={my(v1,v2,v3,v4); v1=vector(nMax,i,0); v1[1]=1; for(i=1,nMax-1,v1[i+1]=v1[i\r+1]+1); v2=vector(nMax,i,0); v2[1]=1; for(i=2,nMax,v2[i]=v1[i]-v1[i-1]); v3=vector(nMax,i,0); for(i=1,3,v3[i]=2^(i-1)); for(i=4,nMax,v3[i]=if(v2[i-1]==1,5,2*v3[i-fibonacci(v1[i-1]+1)]-if(v2[i]==1,1,0))); v4=vector(nMax,i,0); v4[1]=3; for(i=2,nMax,v4[i]=v4[i-1]+v3[i]); v4}

isok(k) = hammingweight(k) == valuation(k, 2) + 2; \\ Michel Marcus, Jul 06 2022
(Python 3.10+)
from itertools import count, islice
def A355489_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n.bit_count()==(n&-n).bit_length()+1,count(max(startvalue,1)))
A355480_list = list(islice(A355489_gen(),30)) # Chai Wah Wu, Jul 15 2022

a(n) = a(n-1) + b(n) for n > 1 with a(1) = 3 where b(n) = {2^(n-1) if n < 4; 5 if c(n-1) = 1; otherwise 2*b(n - A000045(A072649(n-1) + 1)) - [c(n) = 1]} and where c(n) = A010056(n).
A025480(a(n)-1) = A048679(n) for n > 0.
a(A000045(n)) = 2^(n-1) + 1 for n > 1.

A381836 k/25 is in this list if A053824(k) < A112765(k), i.e. if digitsum(k, 5) < valuation(k, 5).

Original entry on oeis.org

1, 5, 10, 25, 30, 50, 75, 125, 130, 150, 175, 250, 275, 375, 500, 625, 630, 650, 675, 750, 775, 875, 1000, 1250, 1275, 1375, 1500, 1875, 2000, 2500, 3125, 3130, 3150, 3175, 3250, 3275, 3375, 3500, 3750, 3775, 3875, 4000, 4375, 4500, 5000, 5625, 6250, 6275, 6375

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A053824, A112765.

Cf. A371176 (base 2), A381838 (base 3), A381837 (base 4).

aList := upto -> local k; [seq(k/25, k in select(n -> add(convert(n, base, 5)) < padic[ordp](n, 5), [seq(25..upto,25)]))]: aList(160000);

Select[Range[160000],DigitSum[#,5]Stefano Spezia, Mar 08 2025 *)

def aList(upto, b): return [n/b^2 for n in srange(b^2, upto, b^2) if sum(n.digits(b)) < valuation(n, b)]
print(aList(160000, 5))

A381837 k/16 is in this list if A053737(k) < A235127(k), i.e. if digitsum(k, 4) < valuation(k, 4).

Original entry on oeis.org

1, 4, 8, 16, 20, 32, 48, 64, 68, 80, 96, 128, 144, 192, 256, 260, 272, 288, 320, 336, 384, 448, 512, 528, 576, 640, 768, 832, 1024, 1028, 1040, 1056, 1088, 1104, 1152, 1216, 1280, 1296, 1344, 1408, 1536, 1600, 1792, 2048, 2064, 2112, 2176, 2304, 2368, 2560, 2816

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A053737, A235127.

Cf. A371176 (base 2), A381838 (base 3), A381836 (base 5).

aList := upto -> local k; [seq(k/16, k in select(n -> add(convert(n, base, 4)) < padic[ordp](n, 4), [seq(16..upto,16)]))]: aList(46000);

Select[Range[46000],DigitSum[#,4]Stefano Spezia, Mar 08 2025 *)

def aList(upto, b): return [n/b^2 for n in srange(b^2, upto, b^2) if sum(n.digits(b)) < valuation(n, b)]
print(aList(46000, 4))

cp's OEIS Frontend

A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

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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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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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A373579 Irregular triangle read by rows where row n lists (in increasing order) the elements of the strong Schreier set encoded by A371176(2*n).

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A373303 First differences of A371176.

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A104287 Decimal expansion of log base phi of 2.

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

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