A053188 -id:A053188 - cp's OEIS frontend

A236564 Difference between 2^(2n-1) and the nearest square.

1, -1, -4, 7, -17, 23, -89, 7, 28, 112, 448, 1792, -4417, 5503, 22012, -4633, -18532, -74128, -296512, 296863, 1187452, -1181833, -4727332, 4817239, 19268956, -17830441, -71321764, 94338007, 377352028, -9092137, -36368548, -145474192, -581896768, -2327587072, -9310348288

Offset: 1

Alex Ratushnyak, Feb 23 2014

The distances of the even powers 2^(2n) to their nearest squares are obviously all zero and therefore skipped.

			a(1) = 2^1 - 1^2 = 1.
a(2) = 2^3 - 3^2 = -1.
a(3) = 2^5 - 6^2 = 32 - 36 = -4.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A053188, A201125, A238454.

A236564 := proc(n)
    local x,sq,lo,hi ;
    x := 2^(2*n-1) ;
    sq := isqrt(x) ;
    lo := sq^2 ;
    hi := (sq+1)^2 ;
    if abs(x-lo) < abs(x-hi) then
        x-lo ;
    else
        x-hi ;
    end if;
end proc: # R. J. Mathar, Mar 13 2014

Table[2^n - Round[Sqrt[2^n]]^2, {n, 1, 79, 2}] (* Alonso del Arte, Feb 23 2014 *)

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
for n in range(47):
    nn = 2**(2*n+1)
    a = isqrt(nn)
    d1 = nn - a*a
    d2 = (a+1)**2 - nn
    if d2 < d1:  d1 = -d2
    print(str(d1), end=',')

If A201125(n) < A238454(n), a(n) = A201125(n), otherwise a(n) = -A238454(n). [Negative terms are for cases where the nearest square is above 2^(2n-1), not below it.] - Antti Karttunen, Feb 27 2014

A359736 Lexicographically earliest sequence of distinct nonnegative integers such that the sequence d(n) = dist(a(n), SQUARES) has the same sequence of digits.

Original entry on oeis.org

0, 10, 1, 2, 6, 42, 20, 7, 11, 4, 56, 3, 5, 21, 30, 43, 12, 31, 14, 8, 13, 9, 29, 19, 15, 18, 22, 17, 24, 32, 72, 26, 28, 90, 23, 91, 35, 109, 37, 41, 48, 73, 27, 34, 50, 57, 38, 40, 33, 47, 71, 51, 62, 55, 66, 89, 112, 16, 79, 39, 130, 63, 46, 44, 65, 25, 135

Offset: 0

M. F. Hasler and Eric Angelini, Jan 12 2023

In the definition, dist(a(n), SQUARES) = A053188(a(n)) is the distance of a(n) from the nearest square. "... has the same digits" means that the concatenation of the terms yields the same string of digits, for the sequence a(.) and the sequence d(.).

Conjectured to be a permutation of the nonnegative integers. The inverse permutation would start (0, 2, 3, 11, 9, 12, 4, 7, 19, 21, 1, 8, 16, 20, 18, 24, ...)

			Below, row "s" lists the closest square to a(n) and row "d" the absolute difference |a(n)-s|. We have the same sequence of digits in rows a (this sequence) and d:
  n :  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...
  a :  0  10   1   2   6  42  20   7  11   4  56   3   5  21  30 ...
  s :  0   9   1   1   4  36  16   9   9   4  49   4   6  25  25 ...
  d :  0   1   0   1   2   6   4   2   2   0   7   1   1   4   5 ...

Eric Angelini, Digit-spines, personal blog "Cinquante signes" on blogspot.com, Jan. 11, 2023.
Eric Angelini, Digit-spines, personal blog "Cinquante signes" on blogspot.com, Jan. 11, 2023. [Cached copy]

Cf. A053188 (distance from the nearest square), A000290 (the squares).

Cf. A359734, A359737 (similar for primes and Fibonacci numbers).

spine(x->x^2, 200) \\ See A359734 for spine()

A080344 Partial sums of A023969.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 29, 30, 31, 32, 33, 34, 35

Offset: 0

N. J. A. Sloane, Mar 20 2003

nonn

Cf. A000194, A023969, A053188, A056847.

[1/2*(n+1-Floor(Sqrt(n+1)+1/2)-Abs(n+1-(Floor(Sqrt(n+1)+1/2))^2)):n in [0..90]]; // Marius A. Burtea, May 09 2019

f(n) = sqrtint(4*n)-2*sqrtint(n); \\ A023969
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, May 10 2019

from math import isqrt
def A080344(n): return n+1-(k:=(m:=isqrt(n+1))+int(n>=m*(m+1)))-abs(n+1-k**2)>>1 # Chai Wah Wu, Jun 05 2025

From Ridouane Oudra, May 11 2019: (Start)
a(n) = (1/2)*(n + 1 - t - abs(n + 1 - t^2)), where t = floor(sqrt(n+1) + 1/2).
a(n) = (1/2)*(n + 1 - A000194(n+1) - abs(n + 1 - A000194(n+1)^2)).
a(n) = (1/2)*(A056847(n+1) - A053188(n+1)). (End)

A301636 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest square of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 2, 4, 2, 4, 1, 1, 4, 2, 4, 9, 4, 2, 4, 1, 1, 5, 4, 4, 5, 5, 2, 0, 2, 5, 1, 1, 5, 8, 5, 1, 1, 5, 2, 0, 0, 2, 8, 10, 4, 2, 4, 5, 1, 1, 1, 1, 5, 10, 8, 5, 5, 9, 4, 2, 4, 4, 2, 4, 9, 5, 5, 8, 10, 9, 5, 5, 9, 9, 5, 5, 9, 4, 2, 4, 10

Offset: 0

Rémy Sigrist, Mar 25 2018

The distance between two Gaussian integers is not necessarily integer, hence the use of the square of the distance.
This sequence is a complex variant of A053188.
See A301626 for the square array dealing with cubes of Gaussian integers.

			Square array begins:
  n\k|    0    1    2    3    4    5    6    7    8    9   10
  ---+-------------------------------------------------------
    0|    0    1    0    1    4    9    4    1    0    1    4
    1|    0    1    1    2    4    5    5    2    1    2    5
    2|    1    2    4    2    1    2    5    5    4    5    8
    3|    1    2    4    1    0    1    4    9    9   10    8
    4|    0    1    4    2    1    2    5   10   16   10    5
    5|    1    2    5    5    4    5    8   10   13    9    4
    6|    4    5    8   10    8    5    4    5    8   10    5
    7|    4    5    8   10    5    2    1    2    5   10    8
    8|    1    2    5    9    4    1    0    1    4    9   13
    9|    0    1    4    9    5    2    1    2    5   10   17
   10|    1    2    5   10    8    5    4    5    8   13   20

Rémy Sigrist, Colored scatterplot for abs(x) <= 500 and abs(y) <= 500 (where the hue is function of sqrt(T(abs(x), abs(y))))
Rémy Sigrist, Voronoi diagram of the squares of Gaussian integers for abs(x) <= 500 and abs(y) <= 500
Rémy Sigrist, Scatterplot of (x, y) such that T(abs(x), abs(y)) is a square and abs(x) <= 500 and abs(y) <= 500
Rémy Sigrist, PARI program for A301636
Wikipedia, Gaussian integer
Wikipedia, Voronoi diagram
Index entries for sequences related to distance to nearest element of some set

Cf. A000290, A001105, A053188, A301626.

PARI
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See Links section.
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T(n, 0) <= A053188(n)^2.
T(n, 0) = 0 iff n is a square (A000290).
T(0, k) = 0 iff k is twice a square (A001105).
T(n, k) = 0 iff n + k*i = z^2 for some Gaussian integer z.

A131866 Distance of n-th semiprime to nearest square.

Original entry on oeis.org

0, 2, 0, 1, 2, 1, 4, 3, 0, 1, 3, 2, 1, 2, 3, 3, 0, 2, 6, 7, 6, 2, 1, 5, 7, 4, 1, 4, 5, 6, 9, 7, 6, 5, 6, 10, 6, 3, 2, 0, 1, 2, 8, 11, 10, 3, 2, 1, 1, 2, 11, 11, 10, 8, 3, 0, 8, 9, 13, 11, 9, 2, 5, 6, 7, 9, 10, 13, 12, 11, 10, 8, 7, 6, 4, 1, 10, 12, 9, 7, 3, 2, 3, 6, 9, 11, 15, 11, 2, 0, 2, 6, 9, 10, 12

Offset: 1

Jonathan Vos Post, Oct 04 2007

This to semiprimes A001358 as A047972 is to primes A000040.

For each semiprime, find the closest square (preceding or succeeding); subtract, take absolute value.

			a(1) = 0 because the first semiprime is 4, which is a square.
a(2) = 2 because the 2nd semiprime is 6 and |6-4| = 2 where 4 is the nearest square to 6.
a(3) = 0 because the 3rd semiprime is 9, which is a square.
a(4) = 1 because the 4th semiprime is 10 and |10-9| = 1 where 9 is the nearest square to 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A047972, A053188.

dns[n_]:=Min[n-Floor[Sqrt[n]]^2,Ceiling[Sqrt[n]]^2-n]; dns/@Select[ Range[ 400],PrimeOmega[#]==2&] (* Harvey P. Dale, Aug 12 2016 *)

a(n)=A053188(A001358(n)) (corrected by R. J. Mathar, Nov 19 2007).

More terms from R. J. Mathar, Oct 24 2007

A133610 Partial sums of pyramidal sequence A053616.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 7, 7, 8, 10, 12, 13, 13, 14, 16, 19, 21, 22, 22, 23, 25, 28, 31, 33, 34, 34, 35, 37, 40, 44, 47, 49, 50, 50, 51, 53, 56, 60, 64, 67, 69, 70, 70, 71, 73, 76, 80, 85, 89, 92, 94, 95, 95, 96, 98, 101, 105, 110, 115, 119, 122, 124, 125, 125, 126, 128, 131, 135

Offset: 0

Jonathan Vos Post, Dec 28 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A002262, A053188, A053616.

a133610 n = a133610_list !! n
a133610_list = scanl1 (+) a053616_list
-- Reinhard Zumkeller, Jan 24 2014

a(n) = SUM[i=0..n] distance from i to nearest triangular number = SUM[i=0..n]MIN{|i - (k*(k+1)/2)|} = SUM[i=0..n] MIN{|A000217(i) - i|}.

Data corrected by Reinhard Zumkeller, Jan 24 2014

A289642 Number of 2-digit numbers whose digits add up to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Miquel Cerda, Jul 09 2017

The 2-digit numbers distributed according to the sum of their digits n.

Symmetrical sequence; a(n) = a(19 - n).

			n(5) = 5 because there are 5 numbers whose digits sum = 5 (14, 23, 32, 41, 50).

Cf. A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278969 (7-digit numbers), A278971 (8-digit numbers), A289354 (9-digit numbers), A053188, A074989, A004739, A066635, A154840, A249121.

G.f.: (1 - x^10)*(x - x^10)/(1 - x)^2.

a(n) = (19-abs(n-9)-abs(n-10))/2 for n=1..18. - Wesley Ivan Hurt, Jul 09 2017

A301297 Distance from n to nearest Catalan number.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33

Offset: 1

N. J. A. Sloane, Mar 24 2018

nonn

Cf. A000108, A053188, A080732, A296239.

A309914 Distance from n to closest triangular number that is different from n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 1, 2, 2, 1, 5, 1, 2, 3, 2, 1, 6, 1, 2, 3, 3, 2, 1, 7, 1, 2, 3, 4, 3, 2, 1, 8, 1, 2, 3, 4, 4, 3, 2, 1, 9, 1, 2, 3, 4, 5, 4, 3, 2, 1, 10, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 11, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 12, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 13, 1, 2, 3, 4, 5, 6, 7, 6, 5

Offset: 0

Ilya Gutkovskiy, Aug 22 2019

nonn

Index entries for sequences related to distance to nearest element of some set

Cf. A000217, A053188, A053616, A066635.

a[n_] := Module[{k = 1}, While[! IntegerQ[Sqrt[8 (n + k) + 1]] && ! IntegerQ[Sqrt[8 (n - k) + 1]], k++]; k]; Table[a[n], {n, 0, 100}]

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A236564 Difference between 2^(2n-1) and the nearest square.

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A359736 Lexicographically earliest sequence of distinct nonnegative integers such that the sequence d(n) = dist(a(n), SQUARES) has the same sequence of digits.

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A080344 Partial sums of A023969.

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A301636 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest square of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

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A131866 Distance of n-th semiprime to nearest square.

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A133610 Partial sums of pyramidal sequence A053616.

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A289642 Number of 2-digit numbers whose digits add up to n.

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A301297 Distance from n to nearest Catalan number.

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