A271410 -id:A271410 - cp's OEIS frontend

A333492 Position of first appearance of n in A271410 (LCM of binary indices).

1, 2, 4, 8, 16, 6, 64, 128, 256, 18, 1024, 12, 4096, 66, 20, 32768, 65536, 258, 262144, 24, 68, 1026, 4194304, 132, 16777216, 4098, 67108864, 72, 268435456, 22, 1073741824, 2147483648, 1028, 65538, 80, 264, 68719476736, 262146, 4100, 144, 1099511627776, 70, 4398046511104

Offset: 1

Gus Wiseman, Mar 28 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The sequence together with the corresponding binary expansions and binary indices begins:
      1:                 1 ~ {1}
      2:                10 ~ {2}
      4:               100 ~ {3}
      8:              1000 ~ {4}
     16:             10000 ~ {5}
      6:               110 ~ {2,3}
     64:           1000000 ~ {7}
    128:          10000000 ~ {8}
    256:         100000000 ~ {9}
     18:             10010 ~ {2,5}
   1024:       10000000000 ~ {11}
     12:              1100 ~ {3,4}
   4096:     1000000000000 ~ {13}
     66:           1000010 ~ {2,7}
     20:             10100 ~ {3,5}
  32768:  1000000000000000 ~ {16}
  65536: 10000000000000000 ~ {17}
    258:         100000010 ~ {2,9}

Giovanni Resta, Table of n, a(n) for n = 1..1000

The version for prime indices is A330225.
The version for standard compositions is A333225.
Let q(k) be the binary indices of k:
- The sum of q(k) is A029931(k).
- The elements of q(k) are row k of A048793.
- The product of q(k) is A096111(k).
- The LCM of q(k) is A271410(k).
- The GCD of q(k) is A326674(k).
GCD of prime indices is A289508.
LCM of prime indices is A290103.
LCM of standard compositions is A333226.
Cf. A000120, A066099, A070939, A074761, A076078, A124767, A285572, A324837, A328219, A328451, A331579, A333227.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[LCM@@bpe[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

Terms a(23) and beyond from Giovanni Resta, Mar 29 2020

A271984 Numbers n such that the denominator of the sum of the reciprocals of the exponents in the binary expansion of 2n is not equal to their LCM. That is, A271410(n) != A116417(n).

Original entry on oeis.org

34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 98, 99, 100, 101, 102, 103, 108, 109, 110, 111, 114, 115, 116, 117, 118, 119, 124, 125, 126, 127, 164, 165, 166, 167, 172, 173, 174, 175, 180, 181, 182, 183, 188, 189, 190, 191

Offset: 1

Peter Kagey, Apr 17 2016

a(2*n) = 1 + a(2*n-1) for all n > 0.

			a(1) = 34 because 34*2 = 68 is the first number such that the LCM of the exponents in its binary expansion (2 and 6) is unequal to the denominator of the sum of reciprocals: lcm(2, 6) = 6 != denominator(1/2 + 1/6) = 3.
Equivalently, A271410(34) = 6 != A116417(34) = 3.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A116417, A271410.

Select[Range@ 1000, (LCM @@ # != Denominator[ Total[1/#]]) &@ Flatten@ Position[ Reverse@ IntegerDigits[#, 2], 1] &] (* Giovanni Resta, Apr 18 2016 *)

A272020 Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 3, 2, 3, 2, 1, 4, 4, 1, 4, 2, 4, 2, 1, 4, 3, 4, 3, 1, 4, 3, 2, 4, 3, 2, 1, 5, 5, 1, 5, 2, 5, 2, 1, 5, 3, 5, 3, 1, 5, 3, 2, 5, 3, 2, 1, 5, 4, 5, 4, 1, 5, 4, 2, 5, 4, 2, 1, 5, 4, 3, 5, 4, 3, 1, 5, 4, 3, 2, 5, 4, 3, 2, 1, 6, 6, 1, 6, 2, 6, 2, 1

Offset: 0

Peter Kagey, Apr 17 2016

Length of n-th row given by A000120(n);
Min of n-th row given by A001511(n);
Sum of n-th row given by A029931(n);
Product of n-th row given by A096111(n);
Max of n-th row given by A113473(n);
Numerator of sum of reciprocals of n-th row given by A116416(n);
Denominator of sum of reciprocals of n-th row given by A116417(n);
LCM of n-th row given by A271410(n).
The first appearance of n is at A001787(n - 1).
n-th row begins at index A000788(n - 1) for n > 0.
Also the reversed positions of 1's in the reversed binary expansion of n. Also the reversed partial sums of the n-th composition in standard order (row n of A066099). Reversing rows gives A048793. - Gus Wiseman, Jan 17 2023

			Row n is given by the exponents in the binary expansion of 2*n. For example, row 5 = [3, 1] because 2*5 = 2^3 + 2^1.
Row 0: []
Row 1: [1]
Row 2: [2]
Row 3: [2, 1]
Row 4: [3]
Row 5: [3, 1]
Row 6: [3, 2]
Row 7: [3, 2, 1]

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000120, A096111, A116416, A116417, A271410, A001787.
Cf. A048793 gives the rows in reverse order.
Cf. A272011.
Lasts are A001511.
Heinz numbers of the rows are A019565.
Firsts are A029837 or A070939 or A113473.
Row sums are A029931.
A066099 lists standard comps, partial sums A358134, weighted sum A359042.
Cf. A005940, A059893, A125106, A161511, A242628.

T:= proc(n) local i, l, m; l:= NULL; m:= n;
      if n=0 then return [][] fi; for i while m>0 do
      if irem(m, 2, 'm')=1 then l:=i, l fi od; l
    end:
seq(T(n), n=0..35);  # Alois P. Heinz, Nov 27 2024

Table[Reverse[Join@@Position[Reverse[IntegerDigits[n,2]],1]],{n,0,100}] (* Gus Wiseman, Jan 17 2023 *)

A333226 Least common multiple of the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 2, 2, 1, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 6, 5, 4, 4, 3, 6, 6, 3, 4, 6, 2, 2, 6, 2, 2, 2, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 7, 6, 10, 5, 12, 4, 4, 4, 12, 3, 6, 6, 3, 6, 6, 3, 10, 4, 6, 6, 6, 2, 2

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

The version for binary indices is A271410.
The version for prime indices is A290103.
Positions of first appearances are A333225.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- The GCD of q(k) is A326674(k).
- The LCM of q(k) is A333226(k).
Cf. A000120, A029931, A048793, A074971, A076078, A233564, A285572, A289508, A289509, A324837, A333227, A333492.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[LCM@@stc[n],{n,100}]

A116417 If n = Sum_{m>=1} 2^(m-1) * b(n,m), where each b(n,m) is 0 or 1 and the sum is a finite sum, then a(n) = denominator of Sum_{m>=1} b(n,m)/m.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 4, 12, 12, 12, 12, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 20, 20, 60, 60, 60, 60, 6, 6, 3, 3, 2, 2, 1, 1, 12, 12, 12, 12, 4, 4, 4, 4, 30, 30, 15, 15, 10, 10, 5, 5, 60, 60, 60, 60, 20, 20, 20, 20, 7, 7, 14, 14, 21, 21, 42, 42, 28, 28, 28, 28, 84, 84, 84, 84

Offset: 0

Leroy Quet, Feb 13 2006

			13 in binary is 1101. So a(13) is the denominator of 1/4 + 1/3 + 1 = 19/12, since the binary digits at positions (from right to left) 1, 3 and 4 are each 1 and the other digits are 0.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A007088, A116416, A271410.

Table[Denominator@ Total@ MapIndexed[#1/ First@ #2 &, Reverse@ IntegerDigits[n, 2]], {n, 0, 79}] (* Michael De Vlieger, Aug 19 2017 *)

a(n) = {my(b = Vecrev(binary(n))); denominator(sum(k=1, #b, b[k]/k));} \\ Michel Marcus, Apr 18 2016

More terms from Joshua Zucker, May 03 2006

A333225 Position of first appearance of n in A333226 (LCMs of compositions in standard order).

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 64, 128, 256, 66, 1024, 68, 4096, 258, 132, 32768, 65536, 1026, 262144, 264, 516, 4098

Offset: 1

Gus Wiseman, Mar 26 2020

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The sequence together with the corresponding compositions begins:
       1: (1)
       2: (2)
       4: (3)
       8: (4)
      16: (5)
      18: (3,2)
      64: (7)
     128: (8)
     256: (9)
      66: (5,2)
    1024: (11)
      68: (4,3)
    4096: (13)
     258: (7,2)
     132: (5,3)
   32768: (16)
   65536: (17)
    1026: (9,2)
  262144: (19)
     264: (5,4)

The version for binary indices is A333492.
The version for prime indices is A330225.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- The GCD of q(k) is A326674(k).
- The LCM of q(k) is A333226(k).
Cf. A000120, A029931, A074971, A076078, A233564, A271410, A289508, A289509, A290103, A333227.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
q=Table[LCM@@stc[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A366027 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, if 2^(d-1) appears in the binary expansion of a(n) then d divides n.

Original entry on oeis.org

1, 2, 4, 3, 16, 5, 64, 8, 256, 17, 1024, 6, 4096, 65, 20, 9, 65536, 7, 262144, 10, 68, 1025, 4194304, 11, 16777216, 4097, 257, 66, 268435456, 18, 1073741824, 128, 1028, 65537, 80, 12, 68719476736, 262145, 4100, 19, 1099511627776, 32, 4398046511104, 1026, 21

Offset: 1

Rémy Sigrist, Sep 26 2023

In other words, the binary expansion of a(n) encodes a subset of the divisors of n.

This sequence is a permutation of the positive integers with inverse A366028.

			The first terms, alongside their binary expansion and the corresponding divisors d, are:
  n   a(n)    bin(a(n))            Corresponding divisors
  --  ------  -------------------  ----------------------
   1       1                    1  {1}
   2       2                   10  {2}
   3       4                  100  {3}
   4       3                   11  {2, 1}
   5      16                10000  {5}
   6       5                  101  {3, 1}
   7      64              1000000  {7}
   8       8                 1000  {4}
   9     256            100000000  {9}
  10      17                10001  {5, 1}
  11    1024          10000000000  {11}
  12       6                  110  {3, 2}
  13    4096        1000000000000  {13}
  14      65              1000001  {7, 1}
  15      20                10100  {5, 3}
  16       9                 1001  {4, 1}
  17   65536    10000000000000000  {17}
  18       7                  111  {3, 2, 1}

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A048793, A271410, A366028 (inverse).

PARI
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See Links section.
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a(p) = 2^(p-1) for any prime number p.

a(2*p) = 2^(p-1) + 1 for any prime number p.

A366028 Inverse permutation to A366027.

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 18, 8, 16, 20, 24, 36, 48, 60, 72, 5, 10, 30, 40, 15, 45, 90, 120, 80, 100, 140, 160, 180, 240, 300, 360, 42, 54, 66, 78, 84, 96, 102, 108, 132, 144, 156, 168, 192, 204, 216, 228, 150, 210, 270, 330, 390, 420, 450, 480, 540, 600, 660, 720

Offset: 1

Rémy Sigrist, Sep 26 2023

For any n > 0, a(n) is a multiple of A271410(n).

			A366027(42) = 32, hence a(32) = 42.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A271410, A366027.

PARI
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See Links section.
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A337744 Numbers of the form Sum_{e in S} 2^(e-1) where S is a finite set of positive integers such that any element of S divides the sum of the elements of S.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 16, 32, 39, 42, 64, 128, 175, 256, 291, 292, 512, 537, 1024, 2048, 2087, 2090, 2181, 2184, 2350, 4096, 8192, 8267, 16384, 16437, 16902, 16912, 32768, 34983, 34986, 65536, 131072, 131342, 131363, 131364, 133127, 133130, 133152, 262144, 524288

Offset: 1

Rémy Sigrist, Sep 26 2020

In other words, this sequence corresponds to the number m such that A271410(m) divides A029931(m).
For any n > 0, A125297(n) gives the number of positive terms < 2^n.
Every power of 2 belongs to the sequence.

			The first terms, alongside their binary representation and corresponding set S, are:
  n   a(n)  bin(a(n))   S
  --  ----  ----------  ------------------
   1     0           0  {}
   2     1           1  {1}
   3     2          10  {2}
   4     4         100  {3}
   5     7         111  {1, 2, 3}
   6     8        1000  {4}
   7    16       10000  {5}
   8    32      100000  {6}
   9    39      100111  {1, 2, 3, 6}
  10    42      101010  {2, 4, 6}
  11    64     1000000  {7}
  12   128    10000000  {8}
  13   175    10101111  {1, 2, 3, 4, 6, 8}
  14   256   100000000  {9}
  15   291   100100011  {1, 2, 6, 9}
  16   292   100100100  {3, 6, 9}

Rémy Sigrist, Table of n, a(n) for n = 1..316
Mathematics Stack Exchange, A finite set of distinct positive numbers is special if each integer in the set divides the sum of all integers within the set.
Index entries for sequences related to binary expansion of n

Cf. A029931, A125297, A271410.

is(n) = { my (b=Vecrev(binary(n)), s=select(k -> b[k], [1..#b])); vecsum(s) % lcm(s)==0 }

A358970 Nonnegative numbers m such that if 2^k appears in the binary expansion of m, then k+1 divides m.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 36, 60, 128, 136, 168, 261, 288, 520, 530, 540, 630, 640, 1056, 2052, 2088, 2100, 2184, 2208, 2304, 2340, 2520, 2580, 4134, 8232, 8400, 8820, 9240, 10248, 10920, 16440, 16560, 16920, 16950, 17010, 17040, 17190, 17280, 18480, 18600, 18720

Offset: 1

Rémy Sigrist, Dec 07 2022

In other words, numbers whose binary expansion encodes a subset of their divisors.
Also numbers m divisible by A271410(m).
This sequence is infinite as it contains A058891.

			60 = 2^5 + 2^4 + 2^3 + 2^2 and 60 is divisible by 5+1, 4+1, 3+1 and 2+1, so 60 belongs to the sequence.
42 = 2^5 + 2^3 + 2^1 and 42 is not divisible by 3+1, so 42 does not belong to the sequence.

Index entries for sequences related to binary expansion of n

Cf. A058891, A271410, A320673.

Select[Range[20000], Function[n, AllTrue[Position[Reverse@ IntegerDigits[n, 2], 1][[All, 1]], Divisible[n, #] &]]] (* Michael De Vlieger, Dec 12 2022 *)

is(n) = { my (r=n, k); while (r, r-=2^k=valuation(r,2); if (n%(k+1), return (0););); return (1); }

def ok(n): return all(n%(k+1) == 0 or not n&(1<Michael S. Branicky, Dec 07 2022

from itertools import count, islice
def A358970_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not any(n%i for i,b in enumerate(bin(n)[:1:-1],1) if b=='1'),count(max(startvalue,0)))
A358970_list = list(islice(A358970_gen(),20)) # Chai Wah Wu, Dec 12 2022

cp's OEIS Frontend

A333492 Position of first appearance of n in A271410 (LCM of binary indices).

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A271984 Numbers n such that the denominator of the sum of the reciprocals of the exponents in the binary expansion of 2n is not equal to their LCM. That is, A271410(n) != A116417(n).

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A272020 Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.

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A333226 Least common multiple of the n-th composition in standard order.

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A116417 If n = Sum_{m>=1} 2^(m-1) * b(n,m), where each b(n,m) is 0 or 1 and the sum is a finite sum, then a(n) = denominator of Sum_{m>=1} b(n,m)/m.

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A333225 Position of first appearance of n in A333226 (LCMs of compositions in standard order).

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A366027 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, if 2^(d-1) appears in the binary expansion of a(n) then d divides n.

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A366028 Inverse permutation to A366027.

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