A257213 -id:A257213 - cp's OEIS frontend

A257217 A257213 - A003059, where A257213(n) = min{d>0 | floor(n/d) = floor(n/(d+1))}, A003059(n) = ceiling(sqrt(n)).

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

Offset: 0

M. F. Hasler, Apr 18 2015

nonn

One has a(n) <= a(n-1) except for n = k^2. The positive jumps occur exactly at the squares, cf. formula.

Cf. A003059, A257213.

f[n_] := Block[{d, k}, Reap@ For[k = 0, k <= n, k++, d = 1; While[Floor[k/d] != Floor[k/(d + 1)], d++]; Sow[d - Ceiling[Sqrt@ k]]] // Flatten // Rest]; f@ 85 (* Michael De Vlieger, Apr 18 2015 *)

PARI
```
A257217(n)=A257213(n)-A003059(n)
```

a(k^2-1) = 0 for k > 1. Proof: For n = k^2-1 = (k-1)(k+1), floor(n/k) = k-1 = n/(k+1) but n/(k-1) = k+1, thus A257213(n) = k = ceiling(sqrt(n)).
A257213(n) >= floor(sqrt(n))+1 = A257213(n+1) >= A257213(n) = ceiling(sqrt(n)), with strict inequality (in the second relation) when n is a square. Therefore a(n) >= 1 for all n = k^2.
a(k^2) >= d when k > d(d-1). Proof: This follows from k^2/(k+d) = k-d+d^2/(k+d), which shows that a(k) >= d when k > d(d-1).

A043548 Least separator of first n Egyptian fractions; i.e., least k for which the integers floor(k/m) for m=1,2,...,n are distinct.

Original entry on oeis.org

1, 1, 2, 6, 9, 16, 20, 30, 42, 49, 64, 81, 90, 110, 132, 156, 169, 196, 225, 256, 272, 306, 342, 380, 420, 441, 484, 529, 576, 625, 650, 702, 756, 812, 870, 930, 961, 1024, 1089, 1156, 1225, 1296, 1332, 1406, 1482, 1560, 1640

Offset: 1

Clark Kimberling

nonn

For n > 1: A257213(a(n)) = n. - Reinhard Zumkeller, Apr 19 2015

After the initial 1, 1, the sequence appears to alternate between runs of pronic numbers and squares with run lengths 2,2,3,3,4,4,... - Charlie Neder, Oct 04 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A257213.

a043548 n = f 1 where
   f k = if distinct $ (map (div k)) [n, n-1 .. 1] then k else f (k + 1)
   distinct [] = True; distinct (u:vs@(v:)) = u /= v && distinct vs
-- Reinhard Zumkeller, Apr 19 2015

a(n)={if(n==1, 1, my(t=sqrtint(4*n-3)); n^2 - n*t + t^2\4)} \\ Andrew Howroyd, Feb 04 2023

a(n) = n^2 + floor(sqrt(n-1))*floor(sqrt(n)+1/2) - n*floor(sqrt(n-1)) - n*floor(sqrt(n)+1/2), for n>1. - Ridouane Oudra, Jun 08 2020

a(n) = n^2 - n*t + floor((t^2)/4), where t = floor(sqrt(4*n-3)) for n>1. - Ridouane Oudra, Jan 24 2023

A053405 Definition: A kara B = C, where C is the least nonnegative integer such that C * B >= A and C * (B-1) < A. a(n) is the smallest k such that n kara k is undefined.

Original entry on oeis.org

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

Offset: 4

Robert Lozyniak (11(AT)onna.com), Jan 08 2000

			7 kara 4 = 2 because 2 * 4 > 7 and 2 * 3 < 7.
7 kara 5 is undefined: 7 kara 5 != 2 because 2 * (5-1) > 7 and 7 kara 5 != 1 because 1 * 5 < 7.
4 kara 3 is undefined: 4 kara 2 = 2; 4 kara 4 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 4..10000

Cf. A053087, A257213.

a053405 n = head [a | a <- [1..], n `kara` a == Nothing] where
   kara a b = if null ks then Nothing else Just $ head ks
              where ks = [c | c <- [1..a], a <= c * b, a > c * (b - 1)]
-- Reinhard Zumkeller, Mar 30 2013

kara[a_, b_] := Module[{r, c}, r = Reduce[c*b >= a && c*(b-1) < a, c, Integers]; If[r === False, Null, c /. {ToRules[r]} // First]]; a[n_] := For[k = 2, True, k++, If[!IntegerQ[n ~kara~ k], Return[k]]]; Table[a[n], {n, 4, 88}] (* Jean-François Alcover, Sep 13 2013 *)

a(n) = A257213(n-1) + 1. - Ridouane Oudra, Apr 19 2025

Corrected and extended by James Sellers, Jan 10 2000

A257212 Least d>0 such that floor(n/d) - floor(n/(d+1)) <= 1.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 18 2015

nonn

An efficient formula for this sequence could be useful for faster computation of A024916.

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

Cf. A257213, A024916.

a257212 n = head [d | d <- [1..], div n d - div n (d+1) <= 1]
-- Reinhard Zumkeller, Apr 19 2015

f[n_] := Block[{d, k}, Reap@ For[k = 0, k <= n, k++, d = 1; While[Floor[k/d] - Floor[k/(d + 1)] > 1, d++]; Sow[d]] // Flatten // Rest]; f@ 86 (* Michael De Vlieger, Apr 18 2015 *)
ld[n_]:=Module[{d=1},While[Floor[n/d]-Floor[n/(d+1)]>1,d++];d]; Array[ ld,90,0] (* Harvey P. Dale, Oct 18 2015 *)

a(n)=for(d=1,n+1,1>=n\d-n\(d+1)&&return(d))

a(n) <= ceiling(sqrt(n)) <= A257213(n) for all n>0.

cp's OEIS Frontend

A257217 A257213 - A003059, where A257213(n) = min{d>0 | floor(n/d) = floor(n/(d+1))}, A003059(n) = ceiling(sqrt(n)).

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A043548 Least separator of first n Egyptian fractions; i.e., least k for which the integers floor(k/m) for m=1,2,...,n are distinct.

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A053405 Definition: A kara B = C, where C is the least nonnegative integer such that C * B >= A and C * (B-1) < A. a(n) is the smallest k such that n kara k is undefined.

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A257212 Least d>0 such that floor(n/d) - floor(n/(d+1)) <= 1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula