

A063655


Smallest semiperimeter of integral rectangle with area n.


21



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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 db) 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 0thick 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(n1).
a(1), a(2) and a(6) are the only unique values  the others appear multiple times.
(End)


LINKS

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


FORMULA

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n1) + A162348(2n).  Daniel Forgues, Sep 29 2014


EXAMPLE

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


MAPLE

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: # Winston C. Yang, Feb 03 2002


MATHEMATICA

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 *)


PROG

(PARI) 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
(Python)
from sympy import divisors
def A063655(n):
d = divisors(n)
l = len(d)
return d[(l1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019


CROSSREFS

Cf. A027709, A033676, A033677, A092510, A162348.
Sequence in context: A071323 A071324 A321441 * A111234 A117248 A079788
Adjacent sequences: A063652 A063653 A063654 * A063656 A063657 A063658


KEYWORD

nonn


AUTHOR

Floor van Lamoen, Jul 24 2001


EXTENSIONS

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


STATUS

approved



