A333225 -id:A333225 - cp's OEIS frontend

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

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}]

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

Original entry on oeis.org

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

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

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

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

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A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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