A162348 -id:A162348 - cp's OEIS frontend

A033676 Largest divisor of n <= sqrt(n).

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

Offset: 1

N. J. A. Sloane

a(n) = sqrt(n) is a new record if and only if n is a square. - Zak Seidov, Jul 17 2009
a(n) = A060775(n) unless n is a square, when a(n) = A033677(n) = sqrt(n) is strictly larger than A060775(n). It would be nice to have an efficient algorithm to calculate these terms when n has a large number of divisors, as for example in A060776, A060777 and related problems such as A182987. - M. F. Hasler, Sep 20 2011
a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Nov 25 2012
a(n) is the smallest central divisor of n. Column 1 of A207375. - Omar E. Pol, Feb 26 2019
a(n^4+n^2+1) = n^2-n+1: suppose that n^2-n+k divides n^4+n^2+1 = (n^2-n+k)*(n^2+n-k+2) - (k-1)*(2*n+1-k) for 2 <= k <= 2*n, then (k-1)*(2*n+1-k) >= n^2-n+k, or n^2 - (2*k-1)*n + (k^2-k+1) = (n-k+1/2)^2 + 3/4 < 0, which is impossible. Hence the next smallest divisor of n^4+n^2+1 than n^2-n+1 is at least n^2-n+(2*n+1) = n^2+n+1 > sqrt(n^4+n^2+1). - Jianing Song, Oct 23 2022

G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A033677 (n/a(n)), A000196 (sqrt), A027750 (list of divisors), A056737 (n/a(n) - a(n)), A219695 (half of this for odd numbers), A207375 (list the central divisor(s)).
The strictly inferior case is A060775. Cf. also A140271.
Indices of given values: A008578 (1 and the prime numbers: a(n) = 1), A161344 (a(n) = 2), A161345 (a(n) = 3), A161424 (4), A161835 (5), A162526 (6), A162527 (7), A162528 (8), A162529 (9), A162530 (10), A162531 (11), A162532 (12), A282668 (indices of primes).

a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Jun 04 2012

A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; end if; end do: a; end proc: # R. J. Mathar, Aug 09 2009

largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* Borislav Stanimirov, Mar 28 2010 *)
Table[Last[Select[Divisors[n],#<=Sqrt[n]&]],{n,100}] (* Harvey P. Dale, Mar 17 2017 *)

A033676(n) = {local(d);if(n<2,1,d=divisors(n);d[(length(d)+1)\2])} \\ Michael B. Porter, Jan 30 2010

from sympy import divisors
def A033676(n):
    d = divisors(n)
    return d[(len(d)-1)//2]  # Chai Wah Wu, Apr 05 2021

a(n) = n / A033677(n).
a(n) = A161906(n,A038548(n)). - Reinhard Zumkeller, Mar 08 2013
a(n) = A162348(2n-1). - Daniel Forgues, Sep 29 2014

A033677 Smallest divisor of n >= sqrt(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 5, 7, 11, 23, 6, 5, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 7, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10

Offset: 1

N. J. A. Sloane

a(n) is the smallest k such that n appears in the k X k multiplication table and A027424(k) is the number of n with a(n) <= k.
a(n) is the largest central divisor of n. Right border of A207375. - Omar E. Pol, Feb 26 2019
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence selects the smallest superior divisor of n. - Gus Wiseman, Feb 19 2021
a(p) = p for p a prime or 1, these are also the record high points in this sequence. - Charles Kusniec, Aug 26 2022
a(n^4+n^2+1) = n^2+n+1 (see A033676). - Jianing Song, Oct 23 2022

			From _Gus Wiseman_, Feb 19 2021: (Start)
The divisors of 36 are {1,2,3,4,6,9,12,18,36}. Of these {1,2,3,4,6} are inferior and {6,9,12,18,36} are superior, so a(36) = 6.
The divisors of 40 are {1,2,4,5,8,10,20,40}. Of these {1,2,4,5} are inferior and {8,10,20,40} are superior, so a(40) = 8.
(End)

G. Tenenbaum, pp. 268ff of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A027424, A056737, A060775, A219695.
The lower central divisor is A033676.
The strictly superior case is A140271.
Leftmost column of A161908 (superior divisors).
Rightmost column of A207375 (central divisors).
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A063538/A063539 list numbers with/without a superior prime divisor.
A070038 adds up superior divisors.
A341676 selects the unique superior prime divisor.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A059172, A116882, A116883, A341591, A341592, A341593, A341675.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001248, A001221, A001222, A020639, A051283.

a033677 n = head $
   dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Oct 20 2011

A033677 := proc(n)
    n/A033676(n) ;
end proc:

Table[Select[Divisors[n], # >= Sqrt[n] &, 1] // First, {n, 80}]  (* Jean-François Alcover, Apr 01 2011 *)

A033677(n) = {local(d); d=divisors(n); d[length(d)\2+1]} \\ Michael B. Porter, Feb 26 2010

from sympy import divisors
def A033677(n):
    d = divisors(n)
    return d[len(d)//2]  # Chai Wah Wu, Apr 05 2021

a(n) = n/A033676(n).

a(n) = A162348(2n). - Daniel Forgues, Sep 29 2014

A063655 Smallest semiperimeter of integral rectangle with area n.

Original entry on oeis.org

2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 14, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 16, 18, 17, 68, 21, 26

Offset: 1

Floor van Lamoen, Jul 24 2001

nonn

Similar to A027709, which is minimal perimeter of polyomino of n cells, or equivalently, minimal perimeter of rectangle of area at least n and with integer sides. Present sequence is minimal semiperimeter of rectangle with area exactly n and with integer sides. - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 03 2002
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n. That is, b = largest divisor of n <= sqrt(n), d = smallest divisor of n >= sqrt(n). a(n) = n+1 iff n is noncomposite (1 or prime). - Daniel Forgues, Nov 22 2009
From Juhani Heino, Feb 05 2019: (Start)
Basis for any thickness "frames" around the minimal area. Perimeter can be thought as the 0-thick frame, it is obviously 2a(n). Thickness 1 is achieved by laying unit tiles around the area, there are 2(a(n)+2) of them. Thickness 2 comes from the second such layer, now there are 4(a(n)+4) and so on. They all depend only on a(n), so they share this structure:
Every n > 1 is included. (For different thicknesses, every integer that can be derived from these with the respective formula. So, the perimeter has every even n > 2.)
For each square n > 1, a(n) = a(n-1).
a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
(End)
Gives a discrete Uncertainty Principle. A complex function on an abelian group of order n and its Discrete Fourier Transform must have at least a(n) nonzero entries between them. This bound is achieved by the indicator function on a subgroup of size closest to sqrt(n). - Oscar Cunningham, Oct 10 2021
Also two times the median divisor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The version for mean instead of median is A057020/A057021. Other doubled medians of multisets are: A360005 (prime indices), A360457 (distinct prime indices), A360458 (distinct prime factors), A360459 (prime factors), A360460 (prime multiplicities), A360555 (0-prepended differences). - Gus Wiseman, Mar 18 2023

			Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Roy Meshulam, An uncertainty inequality for finite abelian groups, arXiv:math/0312407 [math.CO], 2003.
Roy Meshulam, An uncertainty inequality for finite abelian groups, European Journal of Combinatorics, 27 (2006) 63-67.

Cf. A027709, A033676, A033677, A092510, A162348.
Positions of odd terms are A139710.
Positions of even terms are A139711.
A000005 counts divisors, listed by A027750.
A000975 counts subsets with integer median.
Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.

A063655 := proc(n)
    local i,j;
    for i from floor(sqrt(n)) to 1 by -1 do
        j := floor(n/i) ;
        if i*j = n then
            return i+j;
        end if;
    end do:
end proc:
seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002

Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 2*Sqrt[n], d[[len/2]] + d[[1 + len/2]]], {n, 100}] (* T. D. Noe, Mar 06 2012 *)
Table[2*Median[Divisors[n]],{n,100}] (* Gus Wiseman, Mar 18 2023 *)

A063655(n) = { my(c=1); fordiv(n,d,if((d*d)>=n,if((d*d)==n,return(2*d),return(c+d))); c=d); (0); }; \\ Antti Karttunen, Oct 20 2017

from sympy import divisors
def A063655(n):
    d = divisors(n)
    l = len(d)
    return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n-1) + A162348(2n). - Daniel Forgues, Sep 29 2014
a(n) = Min_{d|n} (n/d + d). - Ridouane Oudra, Mar 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001

A162908 Pairs of central divisors of A089229(n).

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 3, 5, 3, 6, 4, 5, 3, 7, 2, 11, 4, 6, 2, 13, 3, 9, 4, 7, 5, 6, 4, 8, 3, 11, 2, 17, 5, 7, 2, 19, 3, 13, 5, 8, 6, 7, 4, 11, 5, 9, 2, 23, 6, 8, 5, 10, 3, 17, 4, 13, 6, 9, 5, 11, 7, 8, 3, 19, 2, 29, 6, 10, 2, 31, 7, 9, 5, 13, 6, 11, 4, 17, 3, 23, 7, 10, 8, 9, 2, 37, 5, 15, 4, 19

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

The central divisors are the two complementary divisors closest to the square

root of a number, as tabulated in A162348.

			a(1)=2 and a(2)=3 since A089229(1)=6 with divisors 1, 2, 3, 4.
a(3)=2 and a(4)=4 since A089229(2)=8 with divisors 1, 2, 4, 8.
a(5)=2 and a(6)=5 since A089229(3)=10 with divisors 1, 2, 5, 10.

Cf. A089229, A162361.

a(2n-1)= A033676(A089229(n)). a(2n)=A033677(A089229(n)). - R. J. Mathar, Jul 19 2009

Missing pairs inserted by R. J. Mathar, Jul 19 2009

cp's OEIS Frontend

A033676 Largest divisor of n <= sqrt(n).

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A033677 Smallest divisor of n >= sqrt(n).

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A063655 Smallest semiperimeter of integral rectangle with area n.

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Maple

Mathematica

PARI

Python

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A162908 Pairs of central divisors of A089229(n).

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