A337223 -id:A337223 - cp's OEIS frontend

A337222 a(n) is the least number that can be obtained by replacing some square XX in the binary expansion of n by X.

0, 1, 2, 1, 2, 5, 2, 3, 4, 5, 2, 5, 4, 5, 6, 3, 4, 9, 10, 9, 4, 5, 10, 11, 8, 9, 6, 11, 12, 13, 6, 7, 8, 9, 18, 17, 4, 9, 18, 19, 8, 9, 10, 11, 20, 5, 22, 11, 12, 17, 18, 19, 12, 13, 6, 23, 24, 25, 14, 27, 12, 13, 14, 7, 8, 17, 18, 19, 34, 17, 34, 35, 8, 9, 18

Offset: 0

Rémy Sigrist, Aug 19 2020

Leading zeros in binary expansions are ignored.

There are four fixed points: 0, 1, 2 and 5; their binary expansion is a squarefree string.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     1      11          1
   4     2     100         10
   5     5     101        101
   6     2     110         10
   7     3     111         11
   8     4    1000        100
   9     5    1001        101
  10     2    1010         10
  11     5    1011        101
  12     4    1100        100
  13     5    1101        101
  14     6    1110        110
  15     3    1111         11
  16     4   10000        100

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A337222
Index entries for sequences related to binary expansion of n

Cf. A020330, A337223, A337224.

PARI
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See Links section.
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a(A020330(n)) = n for any n > 0.

A337224 a(n) is the least number that can be obtained by replacing some repetitive part X^k in the binary expansion of n by X.

Original entry on oeis.org

0, 1, 2, 1, 2, 5, 2, 1, 2, 5, 2, 5, 4, 5, 2, 1, 2, 5, 10, 9, 4, 5, 10, 5, 6, 9, 6, 11, 4, 5, 2, 1, 2, 5, 10, 11, 4, 9, 18, 9, 8, 9, 2, 11, 20, 5, 10, 5, 6, 13, 18, 19, 12, 13, 6, 13, 8, 9, 10, 11, 4, 5, 2, 1, 2, 5, 10, 11, 20, 17, 22, 17, 8, 9, 18, 19, 36, 37

Offset: 0

Rémy Sigrist, Aug 19 2020

Leading zeros in binary expansions are ignored.

There are four fixed points: 0, 1, 2 and 5; their binary expansion is a squarefree string.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     1      11          1
   4     2     100         10
   5     5     101        101
   6     2     110         10
   7     1     111          1
   8     2    1000         10
   9     5    1001        101
  10     2    1010         10
  11     5    1011        101
  12     4    1100        100
  13     5    1101        101
  14     2    1110         10
  15     1    1111          1
  16     2   10000         10

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A337224
Index entries for sequences related to binary expansion of n

Cf. A337222, A337223.

PARI
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See Links section.
```

a(n) = 1 iff n = 2^k-1 for some k > 0.

cp's OEIS Frontend

A337222 a(n) is the least number that can be obtained by replacing some square XX in the binary expansion of n by X.

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