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A063655
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Smallest semiperimeter of integral rectangle with area n.
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37
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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FORMULA
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EXAMPLE
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Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
d = divisors(n)
l = len(d)
return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019
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CROSSREFS
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Positions of odd terms are A139710.
Positions of even terms are A139711.
A000975 counts subsets with integer median.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001
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STATUS
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approved
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