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Search: id:A035106
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| A035106 |
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Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies p_i * p_{i+1}) >= m for some i, 1 <= i <= n-1. |
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12
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| 1, 2, 3, 6, 8, 12, 15, 20, 24, 30, 35, 42, 48, 56, 63, 72, 80, 90, 99, 110, 120, 132, 143, 156, 168, 182, 195, 210, 224, 240, 255, 272, 288, 306, 323, 342, 360, 380, 399, 420, 440, 462, 483, 506, 528, 552, 575, 600, 624, 650, 675, 702, 728, 756, 783, 812, 840
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Equivalently, smallest integer m such that there exists a permutation (p_1, ..., p_n) of (1, ..., n) satisfying p_i * p_{i+1} <= m for every i, 1 <= i <= n-1.
Nonsquare positive integers m such that [sqrt(m)] divides m. Numbers of the form k*(k+1) or k*(k+2). - Max Alekseyev (maxale(AT)gmail.com), Nov 27 2006
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FORMULA
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Theorem: a(n)=n*(n+2)/4 if n is even and (n-1)*(n+3)/4 if n is odd, n>1. - Jud McCranie (j.mccranie(AT)comcast.net), Oct 25 2001
a(n) = a(n-1)+a(n-2)-a(n-3)+1 = A002620(n+2)+A004526(n+2) - Henry Bottomley (se16(AT)btinternet.com), Mar 08 2000
a(n+2) = (2*n^2+12*n+3*(-1)^n+13)/8, i.e. a(n+2) = (n+2)*(n+4)/4 if n is even and (n+1)*(n+5)/4 if n is odd. G.f.: (2-x)/(1-2*x+2*x^3-x^4). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 23 2001
{a[n] == a[n - 2] + (n - 1), a[1] == 0, a[2] == 0}; (-4 - 3*(-1)^n - (-1)^(2*n) + 2*n - 2*(-1)^(2*n)*n + 2*n^2)/8 - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
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EXAMPLE
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n=5: we must arrange the numbers 1..5 so that the max of the products of pairs of adjacent terms is minimized. The answer is 51324, with max product = 8, so a(5) = 8.
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CROSSREFS
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Cf. A064764, A064796, A064797, A064817, A004652, A035104, A035107.
First differences give (essentially) A028242.
Bisections: A002378 (pronic numbers) and A005563.
Cf. A002378, A006446.
Sequence in context: A098393 A103567 A131723 this_sequence A122378 A111242 A133582
Adjacent sequences: A035103 A035104 A035105 this_sequence A035107 A035108 A035109
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), revised Oct 30, 2001
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