A074989 -id:A074989 - cp's OEIS frontend

A053188 Distance from n to nearest square.

0, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4

Offset: 0

Henry Bottomley, Mar 01 2000

			a(7)=2 since 9 is the closest square to 7 and |9-7| = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to distance to nearest element of some set

Cf. A053187, A074989, A000290.

a053188 0 = 0
a053188 n = min (n - last xs) (head ys - n) where
   (xs,ys) = span (< n) a000290_list
-- Reinhard Zumkeller, Nov 28 2011

Flatten[Table[Abs[Nearest[Range[0,25]^2,n]-n],{n,0,120}]]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=abs(((sqrtint(4*n) + 1)\2)^2 - n) \\ Charles R Greathouse IV, Nov 16 2022

from math import isqrt
def A053188(n): return abs(((m:=isqrt(n))+int(n-m*(m+1)>=1))**2-n) # Chai Wah Wu, Aug 03 2022

a(n) = |floor(sqrt(n) + 1/2)^2 - n|. - Ridouane Oudra, May 01 2019

a(n) <= sqrt(n). - Charles R Greathouse IV, Nov 16 2022

A201053 Nearest cube.

Original entry on oeis.org

0, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Reinhard Zumkeller, Nov 28 2011

a(n) = if n-A048763(n) < A048762(n)-n then A048762(n) else A048763(n);

apart from 0, k^3 occurs 3*n^2+1 times, cf. A056107.

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

Cf. A061023, A074989, A053187 (nearest square), A000578.

Cf. A048762, A048763, A056107.

a201053 n = a201053_list !! n
a201053_list = 0 : concatMap (\x -> replicate (a056107 x) (x ^ 3)) [1..]

seq(k^3 $ (3*k^2+1), k=0..10); # Robert Israel, Jan 03 2017

Module[{nn=70,c},c=Range[0,Ceiling[Surd[nn,3]]]^3;Flatten[Array[ Nearest[ c,#]&,nn,0]]] (* Harvey P. Dale, May 27 2014 *)

from sympy import integer_nthroot
def A201053(n):
    a = integer_nthroot(n,3)[0]
    return a**3 if 2*n < a**3+(a+1)**3 else (a+1)**3 # Chai Wah Wu, Mar 31 2021

G.f.: (1-x)^(-1)*Sum_{k>=0} (3*k^2+3*k+1)*x^((k+1)*(k^2+k/2+1)). - Robert Israel, Jan 03 2017

Sum_{n>=1} 1/a(n)^2 = Pi^4/30 + Pi^6/945. - Amiram Eldar, Aug 15 2022

A296239 a(n) = distance from n to nearest Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Rémy Sigrist, Dec 09 2017

The Fibonacci numbers correspond to sequence A000045.
This sequence is analogous to:
- A051699 (distance to nearest prime),
- A053188 (distance to nearest square),
- A053646 (distance to nearest power of 2),
- A053615 (distance to nearest oblong number),
- A053616 (distance to nearest triangular number),
- A061670 (distance to nearest power),
- A074989 (distance to nearest cube),
- A081134 (distance to nearest power of 3),
The local maxima of the sequence correspond to positive terms of A004695.
a(n) = 0 iff n = A000045(k) for some k >= 0.
a(n) = 1 iff n = A061489(k) for some k > 4.
For any n >= 0, abs(a(n+1) - a(n)) <= 1.
For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

			For n = 42:
- A000045(9) = 34 <= 42 <= 55 = A000045(10),
- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* Robert G. Wilson v, Dec 11 2017 *)
With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

A154840 Distance to nearest cube different from n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 2, 1, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

R. J. Mathar, Nov 01 2009

Equals A074989(n) if this is not zero, else 1+A055400(n-1), the distance to the nearest cube < n.

			a(8)=7, because the two cubes below and above 8 are 1^3=1 and 3^3=27, and the distance to 1 is smaller, namely 8-1=7.

Cf. A066635, A163497.

distNearstDiffCub := proc(n) local iscbr ; iroot(n,3,'iscbr') ; if iscbr then 1+A055400(n-1); else A074989(n) ; end if; end proc;

dnc[n_]:=Module[{c=Surd[n,3]},If[IntegerQ[c],n-(c-1)^3,Min[n-Floor[ c]^3, Ceiling[c]^3-n]]]; Array[dnc,90,0] (* Harvey P. Dale, Mar 30 2019 *)

A163497 Numbers n with following property: let c = nearest cube to n that is different from n and let p = nearest prime to n that is different from n. Then |n-c| = |n-p|.

Original entry on oeis.org

2, 25, 28, 119, 126, 340, 345, 728, 731, 1329, 1346, 2188, 2200, 3374, 3382, 4911, 4916, 6858, 6861, 9259, 9269, 12165, 12182, 15622, 15627, 19682, 19685, 24384, 24390, 29790, 29797, 35935, 35944, 42869, 42887, 50652, 50662, 59300, 59326

Offset: 1

Gaurav Kumar, Jul 29 2009

nonn

With the exception of 2 those k where A051699(k) = A074989(k) (same distance to nearest prime and to nearest cube). - R. J. Mathar, Aug 08 2009

			a(1) = 2 since 2 lies between 1 (cube) and 3 (prime);
a(2) = 28 since 28 lies between 27 (cube) and 29 (prime).

Cf. A154840.

A163497 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A051699(a) = A074989(a) then return a; end if; end do ; end if; end proc: # R. J. Mathar, Nov 01 2009

Edited by Zak Seidov, Aug 01 2009

Further edited by N. J. A. Sloane, Oct 31 2009

A301626 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest cube of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 1, 1, 4, 8, 2, 0, 2, 8, 9, 5, 1, 1, 5, 9, 4, 10, 4, 2, 4, 10, 4, 1, 5, 9, 5, 5, 9, 5, 1, 0, 2, 8, 10, 8, 10, 8, 2, 0, 1, 1, 5, 13, 13, 13, 13, 5, 1, 1, 4, 2, 4, 10, 20, 18, 20, 10, 4, 2, 4, 4, 2, 4, 9, 17, 25, 25, 17, 9, 4, 2, 4, 5, 1, 1

Offset: 0

Rémy Sigrist, Mar 24 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 A074989.
See A301636 for the square array dealing with squares of Gaussian integers.

			Square array begins:
  n\k|    0    1    2    3    4    5    6    7    8    9   10
  ---+-------------------------------------------------------
    0|    0    0    1    4    8    9    4    1    0    1    4  -->  A301639
    1|    0    1    1    2    5   10    5    2    1    2    2
    2|    1    1    0    1    4    9    8    5    4    4    1
    3|    4    2    1    2    5   10   13   10    9    5    2
    4|    8    5    4    5    8   13   20   17   13    8    5
    5|    9   10    9   10   13   18   25   25   18   13   10
    6|    4    5    8   13   20   25   32   32   25   20   17
    7|    1    2    5   10   17   25   32   41   34   29   26
    8|    0    1    4    9   13   18   25   34   45   40   37
    9|    1    2    4    5    8   13   20   29   40   53   50
   10|    4    2    1    2    5   10   17   26   37   50   65

Rémy Sigrist, Table of n, a(n) for n = 0..20300
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 cubes 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 A301626
Wikipedia, Gaussian integer
Wikipedia, Voronoi diagram
Index entries for sequences related to distance to nearest element of some set

Cf. A000578, A074989, A301636, A301639 (first row/column).

PARI
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See Links section.
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T(n, k) = T(k, n).
T(n, 0) <= A074989(n)^2.
T(n, 0) = 0 iff n is a cube (A000578).
T(n, k) = 0 iff n + k*i = z^3 for some Gaussian integer z.

A301639 a(n) = square of the distance from n to nearest cube of a Gaussian integer.

Original entry on oeis.org

0, 0, 1, 4, 8, 9, 4, 1, 0, 1, 4, 4, 5, 8, 13, 20, 29, 40, 53, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 130, 117, 106, 97, 90, 85, 82, 81, 82, 85, 90, 97, 106, 117, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25

Offset: 0

Rémy Sigrist, Mar 25 2018

nonn

The distance between two Gaussian integers is not necessarily integer, hence the use of the square of the distance.

This sequence is a variant of A074989: here we minimize norm(n - z^3) where z runs through every Gaussian integers, there we minimize abs(n - m^3) where m runs through every integers.

			For n = 4: the nearest Gaussian cubes to 4 are 2 + 2*i and 2 - 2*i, hence a(4) = (4-2)^2 + 2^2 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of this sequence versus A074989^2 for n = 0..1000
Wikipedia, Gaussian integer
Index entries for sequences related to distance to nearest element of some set

Cf. A074989, A301626.

a(n) = A301626(n, 0).

a(n) <= A074989(n)^2.

A342873 Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).

Original entry on oeis.org

0, 7, 16, 62, 92, 213, 276, 508, 616, 995, 1160, 1722, 1956, 2737, 3052, 4088, 4496, 5823, 6336, 7990, 8620, 10637, 11396, 13812, 14712, 17563, 18616, 21938, 23156, 26985, 28380, 32752, 34336, 39287, 41072, 46638, 48636, 54853, 57076, 63980, 66440, 74067

Offset: 1

Lamine Ngom, Mar 28 2021

That is, numbers k such that A074989(k) = A342872(k).
They form 2 partitions:
7, 62, 213, ... = 8*k^3 - k = k*A157914(k).
0, 16, 92,  ... = 8*k^3 + 6*k^2 + 2*k = 2*k*A033951(k).

Cf. A074989, A342872.

Cf. A157914, A033951, A074378, A201053, A000578, A007531, A081134.

def aupto(limit):
  cubes = [k**3 for k in range(int((limit+1)**1/3)+2)]
  proms = [k*(k+1)*(k+2) for k in range(int((limit+1)**1/3)+1)]
  A074989 = [min(abs(n-c) for c in cubes) for n in range(limit+1)]
  A342872 = [min(abs(n-p) for p in proms) for n in range(limit+1)]
  return [m for m in range(limit+1) if A074989[m] == A342872[m]]
print(aupto(10**4)) # Michael S. Branicky, Mar 28 2021

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

cp's OEIS Frontend

A053188 Distance from n to nearest square.

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A201053 Nearest cube.

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A296239 a(n) = distance from n to nearest Fibonacci number.

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A154840 Distance to nearest cube different from n.

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A163497 Numbers n with following property: let c = nearest cube to n that is different from n and let p = nearest prime to n that is different from n. Then |n-c| = |n-p|.

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A301626 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest cube of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).

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A301639 a(n) = square of the distance from n to nearest cube of a Gaussian integer.

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A342873 Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).