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A063655
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Smallest semiperimeter of integral rectangle with area n.
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10
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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, 17, 72, 17, 74, 39
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Contribution from Daniel Forgues (squid(AT)zensearch.com), Nov 22 2009: (Start)
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n.
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.) (End)
Contribution from Daniel Forgues (squid(AT)zensearch.com), Nov 29 2009: (Start)
If a(n) is not n+1 (n is not 1 or prime) then:
b+d = a(n) is known, bd = n is known, so we have:
(x-b)(x-d) = x^2-(b+d)+bd fully determined and finding the roots
of the quadratic equation gives back the squarest 2-factorization of n. (End)
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EXAMPLE
| 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
| interface(quiet=true); for n from 1 to 100 do for i from floor(sqrt(n)) to 1 by -1 do j := floor(n / i); if (i * j = n) then printf("%d, ", i + j); break; end if; od; od; # from Winston C. Yang
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CROSSREFS
| 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
a(n) = A033676(n) + A033677(n).
Sequence in context: A158973 A071323 A071324 * A117248 A079788 A146288
Adjacent sequences: A063652 A063653 A063654 * A063656 A063657 A063658
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KEYWORD
| nonn
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AUTHOR
| Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 24 2001
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EXTENSIONS
| Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001
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