A272679 -id:A272679 - cp's OEIS frontend

A272680 Smallest square that begins with n (in binary).

0, 1, 4, 25, 4, 81, 25, 121, 16, 9, 81, 361, 25, 441, 225, 121, 16, 1089, 36, 625, 81, 169, 361, 1521, 49, 25, 841, 441, 225, 3721, 121, 2025, 64, 529, 1089, 4489, 36, 2401, 1225, 625, 81, 5329, 169, 2809, 11449, 361, 5929, 1521, 6241, 49, 100, 6561, 841, 6889

Offset: 0

N. J. A. Sloane, May 22 2016

			a(10)=81, because 81 = 9^2 = 1010001_2 begins with 1010 = 10_2.

Allan C. Wechsler, posting to math-fun mailing list May 22 2016.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007088, A018851, A018796, A272679, A272681.

from gmpy2 import isqrt
def A272680(n):
    if n == 0:
        return 0
    else:
        d, nd = 1, n
        while True:
            x = (isqrt(nd-1)+1)**2
            if x < nd+d:
                return int(x)
            d *= 2
            nd *= 2 # Chai Wah Wu, May 22 2016

More terms from Chai Wah Wu, May 22 2016

A272681 Smallest binary square that begins with the binary expansion of n.

Original entry on oeis.org

0, 1, 100, 11001, 100, 1010001, 11001, 1111001, 10000, 1001, 1010001, 101101001, 11001, 110111001, 11100001, 1111001, 10000, 10001000001, 100100, 1001110001, 1010001, 10101001, 101101001, 10111110001, 110001, 11001, 1101001001, 110111001, 11100001, 111010001001

Offset: 0

N. J. A. Sloane, May 22 2016

			a(10)=1010001 = 81_10, because 1010001_2 begins with 1010 = 10_2.

Allan C. Wechsler, posting to math-fun mailing list May 22 2016.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007088, A018851, A018796, A272679, A272680.

from gmpy2 import isqrt
def A272681(n):
    if n == 0:
        return 0
    else:
        d, nd = 1, n
        while True:
            x = (isqrt(nd-1)+1)**2
            if x < nd+d:
                return int(bin(x)[2:])
            d *= 2
            nd *= 2 # Chai Wah Wu, May 22 2016

More terms from Chai Wah Wu, May 22 2016

A319268 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary representation of n^2 starts with the binary representation of a(n).

Original entry on oeis.org

1, 2, 4, 8, 3, 9, 6, 16, 5, 12, 7, 18, 10, 24, 14, 32, 36, 20, 11, 25, 13, 15, 33, 72, 19, 21, 22, 49, 26, 28, 30, 64, 17, 144, 38, 40, 42, 45, 23, 50, 52, 27, 57, 60, 31, 66, 34, 288, 37, 39, 81, 84, 43, 91, 47, 98, 101, 105, 54, 56, 29, 120, 62, 128, 132, 68

Offset: 1

Rémy Sigrist, Sep 16 2018

This sequence is a permutation of the natural numbers with inverse A319499.
We can build a variant of this sequence for any base b > 1.
We can build a variant of this sequence for any strictly increasing sequence of nonnegative integers.

			The first terms, alongside the binary representation of n^2 with a(n) in parentheses, are:
  n   a(n)  bin(n^2)
  --  ----  --------
   1     1          (1)
   2     2        (10)0
   3     4       (100)1
   4     8      (1000)0
   5     3      (11)001
   6     9     (1001)00
   7     6     (110)001
   8    16    (10000)00
   9     5    (101)0001
  10    12    (1100)100
  11     7    (111)1001
  12    18   (10010)000
  13    10   (1010)1001
  14    24   (11000)100
  15    14   (1110)0001
  16    32  (100000)000
  17    36  (100100)001
  18    20  (10100)0100
  19    11  (1011)01001
  20    25  (11001)0000

Ivan Neretin, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A319268
Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of A070939(n^2) - A070939(a(n)))
Index entries for sequences that are permutations of the natural numbers

Cf. A000290, A070939, A272679, A319499.

a = {1}; Do[r = IntegerDigits[n^2, 2]; AppendTo[a, Min@Complement[Table[FromDigits[Take[r, k], 2], {k, Length@r}],a]], {n, 2, 66}]; a (* Ivan Neretin, Oct 24 2018 *)

PARI
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See Links section.
```

A296616 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the binary expansion of a(n) * a(n + 1) starts with the binary expansion of n.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 14, 8, 16, 9, 18, 5, 10, 11, 21, 12, 22, 13, 23, 27, 24, 28, 26, 29, 53, 31, 54, 32, 56, 17, 57, 35, 15, 36, 61, 37, 63, 19, 64, 39, 33, 20, 34, 41, 69, 42, 71, 43, 72, 44, 73, 45, 74, 46, 38, 47, 77, 48, 78, 49, 79, 25, 40, 51, 81, 52, 82

Offset: 1

Rémy Sigrist, Dec 17 2017

It is likely that this sequence is a permutation of the natural numbers.
The lines visible in the scatterplot of the first terms seems to corresponds to set of indices n where the function f(n) = Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)) has the same value; those lines can be partitioned into two groups, depending on the parity of n (see Links section).
This sequence has connections with A272679: here the binary expansion of a(n)*a(n+1) starts with that of n, there the binary expansion of a(n)^2 starts with that of n.

			The first terms, alongside the binary representations of n and a(n) * a(n + 1), are:
  n     a(n)    bin(n)    bin(a(n)*a(n+1))
  --    ----    ------    ----------------
   1       1         1            10
   2       2        10          1000
   3       4        11          1100
   4       3       100         10010
   5       6       101        101010
   6       7       110       1100010
   7      14       111       1110000
   8       8      1000      10000000
   9      16      1001      10010000
  10       9      1010      10100010
  11      18      1011       1011010
  12       5      1100        110010
  13      10      1101       1101110
  14      11      1110      11100111
  15      21      1111      11111100
  16      12     10000     100001000
  17      22     10001     100011110
  18      13     10010     100101011
  19      23     10011    1001101101
  20      27     10100    1010001000

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A296616
Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)))
Rémy Sigrist, Colored scatterplot of the first 10000 terms (where the color is function of the parity of n)

Cf. A029837, A272679.

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A272680 Smallest square that begins with n (in binary).

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A272681 Smallest binary square that begins with the binary expansion of n.

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A319268 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary representation of n^2 starts with the binary representation of a(n).

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A296616 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the binary expansion of a(n) * a(n + 1) starts with the binary expansion of n.

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Crossrefs