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%I A063655 #63 Mar 17 2024 15:59:09
%S A063655 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,
%T A063655 30,11,32,12,14,19,12,12,38,21,16,13,42,13,44,15,14,25,48,14,14,15,20,
%U A063655 17,54,15,16,15,22,31,60,16,62,33,16,16,18,17,68,21,26
%N A063655 Smallest semiperimeter of integral rectangle with area n.
%C A063655 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
%C A063655 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
%C A063655 From _Juhani Heino_, Feb 05 2019: (Start)
%C A063655 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:
%C A063655 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.)
%C A063655 For each square n > 1, a(n) = a(n-1).
%C A063655 a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
%C A063655 (End)
%C A063655 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
%C A063655 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
%H A063655 T. D. Noe, <a href="/A063655/b063655.txt">Table of n, a(n) for n = 1..10000</a>
%H A063655 Roy Meshulam, <a href="https://arxiv.org/abs/math/0312407">An uncertainty inequality for finite abelian groups</a>, arXiv:math/0312407 [math.CO], 2003.
%H A063655 Roy Meshulam, <a href="https://doi.org/10.1016/j.ejc.2004.07.009">An uncertainty inequality for finite abelian groups</a>, European Journal of Combinatorics, 27 (2006) 63-67.
%F A063655 a(n) = A033676(n) + A033677(n).
%F A063655 a(n) = A162348(2n-1) + A162348(2n). - _Daniel Forgues_, Sep 29 2014
%F A063655 a(n) = Min_{d|n} (n/d + d). - _Ridouane Oudra_, Mar 17 2024
%e A063655 Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.
%p A063655 A063655 := proc(n)
%p A063655     local i,j;
%p A063655     for i from floor(sqrt(n)) to 1 by -1 do
%p A063655         j := floor(n/i) ;
%p A063655         if i*j = n then
%p A063655             return i+j;
%p A063655         end if;
%p A063655     end do:
%p A063655 end proc:
%p A063655 seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002
%t A063655 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 *)
%t A063655 Table[2*Median[Divisors[n]],{n,100}] (* _Gus Wiseman_, Mar 18 2023 *)
%o A063655 (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
%o A063655 (Python)
%o A063655 from sympy import divisors
%o A063655 def A063655(n):
%o A063655     d = divisors(n)
%o A063655     l = len(d)
%o A063655     return d[(l-1)//2] + d[l//2] # _Chai Wah Wu_, Jun 14 2019
%Y A063655 Cf. A027709, A033676, A033677, A092510, A162348.
%Y A063655 Positions of odd terms are A139710.
%Y A063655 Positions of even terms are A139711.
%Y A063655 A000005 counts divisors, listed by A027750.
%Y A063655 A000975 counts subsets with integer median.
%Y A063655 Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.
%K A063655 nonn
%O A063655 1,1
%A A063655 _Floor van Lamoen_, Jul 24 2001
%E A063655 Corrected and extended by Larry Reeves (larryr(AT)acm.org) and _Dean Hickerson_, Jul 26 2001

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