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 A063655 Smallest semiperimeter of integral rectangle with area n. 22
 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 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 LINKS 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. FORMULA a(n) = A033676(n) + A033677(n). a(n) = A162348(2n-1) + 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: seq(A063655(n), n=1..80); # 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[(l-1)//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 A343292 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

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Last modified July 1 18:48 EDT 2022. Contains 354974 sequences. (Running on oeis4.)