A018851 -id:A018851 - cp's OEIS frontend

A272679 a(n)^2 is the smallest square whose binary expansion begins with the binary expansion of n.

0, 1, 2, 5, 2, 9, 5, 11, 4, 3, 9, 19, 5, 21, 15, 11, 4, 33, 6, 25, 9, 13, 19, 39, 7, 5, 29, 21, 15, 61, 11, 45, 8, 23, 33, 67, 6, 49, 35, 25, 9, 73, 13, 53, 107, 19, 77, 39, 79, 7, 10, 81, 29, 83, 59, 21, 15, 43, 61, 87, 11, 89, 63, 45, 8, 129, 23, 93, 33, 47

Offset: 0

N. J. A. Sloane, May 22 2016

			a(10)=9, because 9^2 = 81 = 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, A272680, A272681.

from gmpy2 import isqrt
def A272679(n):
    if n == 0:
        return 0
    else:
        d, nd = 1, n
        while True:
            x = isqrt(nd-1)+1
            if x**2 < nd+d:
                return int(x)
            d *= 2
            nd *= 2 # Chai Wah Wu, May 22 2016

More terms from Chai Wah Wu, May 22 2016

A038690 a(n)^2 is smallest square containing the string 'n'.

Original entry on oeis.org

0, 1, 5, 6, 2, 5, 4, 24, 9, 3, 10, 34, 11, 37, 12, 34, 4, 42, 43, 14, 45, 11, 15, 48, 18, 5, 51, 52, 17, 23, 48, 56, 18, 58, 59, 66, 6, 61, 62, 63, 20, 21, 65, 66, 12, 116, 68, 69, 22, 7, 50, 72, 23, 73, 74, 166, 16, 24, 126, 77, 40, 19, 25, 128, 8, 81, 108, 26, 41, 13, 52, 131

Offset: 0

Hans Havermann

"Containment" implies here that the digits of n are consecutive digits in the square; see A091873 for a relaxed alternative. [R. J. Mathar, Dec 09 2008]

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A018851, A029944, A068165, A091873.

Table[ i=0; While[ StringPosition[ ToString[ i^2 ], ToString[ n ] ]=={}, i++ ]; i, {n, 0, 80} ]

A272677 Number of digits in A018796 - number of digits in n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 2, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 0, 2, 2, 1, 2, 2, 3, 2, 1, 3, 2, 2, 3, 1, 3, 0, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 2, 3

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

nonn

These are the numbers d mentioned in Robert Israel's comment in A018851.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A018851, A018796.

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

Original entry on oeis.org

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

A308055 a(1)=2; for n > 1, a(n) is the square root of the smallest square with a(n-1) as a prefix in base 10.

Original entry on oeis.org

2, 5, 23, 48, 22, 15, 39, 63, 251, 501, 224, 474, 689, 2625, 5124, 22637, 47579, 68978, 83053, 28819, 53684, 2317, 4814, 6939, 26342, 51325, 71642, 84642, 290933, 1705676, 4129984, 20322363, 45080332, 67141889, 25911752, 16097128, 12687446, 35619442, 59682026

Offset: 1

Sean Lipton, May 09 2019

The sequence contains the following squares: 315844, 289, 81, 9, and is eventually periodic. - Rémy Sigrist, Jul 13 2019

			a(1) = 2;
25 is the smallest square with 2 as a prefix;
a(2) = sqrt(25) = 5;
529 is the smallest square with 5 as a prefix;
a(3) = sqrt(529) = 23;
2304 is the smallest square with 23 as a prefix;
a(4) = sqrt(2304) = 48; etc.

Rémy Sigrist, Table of n, a(n) for n = 1..100
Rémy Sigrist, PARI program for A308055

Cf. A000290 (squares), A018851.

PARI
```
\\ See Links section.
```

From Rémy Sigrist, Jul 13 2019: (Start)
a(n+1) = A018851(a(n)).
a(n + 4) = a(n) for n >= 83. (End)

a(31)-a(39) from Jon E. Schoenfield, May 10 2019

Comments corrected by Rémy Sigrist, Jul 13 2019

A272676 Smallest number greater than previous term such that a(n)^2 begins with n.

Original entry on oeis.org

0, 1, 5, 6, 7, 23, 25, 27, 29, 30, 32, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 188, 190, 193, 195, 198, 200, 203, 205, 208

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

			a(4) = 7 because 7^2 = 49 begins with 4. (Without the monotonic condition, we could have used 4 - this would give A018851.)

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

Cf. A077502 (the squares), A018851, A018796.

A272678 Smallest number m such that A272677(m) = n.

Original entry on oeis.org

0, 2, 5, 35, 296, 2600, 25317, 251416, 2504474, 25010000, 250044723, 2500100000, 25000316228, 250002000003, 2500004472137, 25000010000000, 250000044721361, 2500000141421358, 25000000316227767

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

Given n, this is the smallest number m with the property that the smallest square beginning with m has n more digits than n.
a(n) >= 25*10^(n-3). Conjecture: a(n)/(25*10^(n-3)) -> 1 as n -> oo. - Chai Wah Wu, May 21 2016
For odd n > 2, it seems that a(n) is about 25 * 10^(n-3) + 10^(floor((n-1)/2)), although a(13) breaks that pattern. - David A. Corneth, May 22 2016
Except for n = 1 and 13, a(n) appears to be approximately equal to either 25*10^(n-3)+sqrt(10^(n-1)) (for n = 0, 2, 3, 5, 6, 9, 11, 12, 15, 18, ... ) or 25*10^(n-3)+sqrt(2*10^(n-1)) (for n = 4, 7, 8, 14, 16, 17, ...). For n = 1, a(n) is approximately 25*10^(n-3)+sqrt(3*10^(n-1)) and for n = 13, a(n) is about equal to 25*10^(n-3)+sqrt(4*10^(n-1)). Conjecture: a(n) is always  approximately to 25*10^(n-3)+sqrt(k*10^(n-1)) for some small integer k > 0. - Chai Wah Wu, May 22 2016
Using the above conjecture as a guide, upper bounds for a(n) can be computed (see file in links) which coincide with a(n) for n <= 19. - Chai Wah Wu, May 23 2016

			The smallest square beginning with 5 is 529, which has two more digits than 5, and corresponds to a(2) = 5.

Chai Wah Wu, Upper bounds for a(n), n = 0..1000

Cf. A018851, A018796, A272677.

a(6)-a(8) from Chai Wah Wu, May 21 2016
a(9)-a(10), a(15)-a(18) and corrected a(12) from Chai Wah Wu, May 22 2016
a(11)-a(14) from David A. Corneth, May 22 2016

A260463 a(n) is the smallest number not already in the sequence such that a(n)^2 begins with n.

Original entry on oeis.org

1, 5, 6, 2, 23, 8, 27, 9, 3, 10, 34, 11, 37, 12, 39, 4, 42, 43, 14, 45, 46, 15, 48, 49, 16, 51, 52, 17, 54, 55, 56, 18, 58, 59, 188, 19, 61, 62, 63, 20, 203, 65, 66, 21, 213, 68, 69, 22, 7, 71, 72, 229, 73, 74, 235, 75, 24, 241, 77, 78, 247, 25, 251, 80, 81, 257, 26, 83, 263, 84, 267, 85, 86, 273, 87, 276, 88, 28, 89

Offset: 1

Derek Orr, Jul 26 2015

Conjectured to be a permutation of the natural numbers.

Differs from A018851 at n = 25.

Derek Orr, Table of n, a(n) for n = 1..10000

Cf. A018851.

v=[];k=1;while(#v<100,d=digits(k^2);D=digits(#v+1);if(#D<=#d,c=1;for(i=1,#D,if(D[i]!=d[i],c=0;break));if(c&&!vecsearch(vecsort(v),k),v=concat(v,k);k=0));k++);v

a(n) >= sqrt(n) for all n > 0. If a(n) = sqrt(n), then n is a square. Note the converse is false: a(25) = 16.

A119764 Smallest number whose square begins with n in binary.

Original entry on oeis.org

1, 2, 5, 2, 9, 5, 15, 4, 3, 9, 19, 5, 21, 15, 11, 4, 33, 6, 25, 9, 13, 19, 39, 7, 5, 29, 21, 15, 61, 11, 45, 8, 65, 33, 67, 6, 49, 35, 25, 9, 73, 13, 53, 106, 19, 77, 39, 79, 7, 10, 81, 29, 83, 59, 21, 15, 43, 61, 87, 11, 89, 63, 45, 8, 23, 65, 93, 33, 47, 67, 135, 12, 97, 69, 49

Offset: 1

Franklin T. Adams-Watters, Aug 04 2006

Binary version of A018851.

cp's OEIS Frontend

A272679 a(n)^2 is the smallest square whose binary expansion begins with the binary expansion of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Extensions

A038690 a(n)^2 is smallest square containing the string 'n'.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A272677 Number of digits in A018796 - number of digits in n.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Extensions

A308055 a(1)=2; for n > 1, a(n) is the square root of the smallest square with a(n-1) as a prefix in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A272676 Smallest number greater than previous term such that a(n)^2 begins with n.

Views

Author

Keywords

Examples

Links

Crossrefs

A272678 Smallest number m such that A272677(m) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A260463 a(n) is the smallest number not already in the sequence such that a(n)^2 begins with n.

Views