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 A182258 Least number k such that there exists a simple graph on k vertices having precisely n spanning trees. 1
 3, 4, 5, 6, 7, 4, 5, 10, 5, 5, 13, 6, 6, 4, 7, 8, 7, 5, 5, 22, 8, 5, 9, 8, 7, 6, 6, 6, 9, 6, 7, 10, 6, 6, 7, 10, 7, 5, 7, 8, 7, 7, 5, 7, 11, 6, 7, 7, 7, 6, 8, 6, 6, 6, 8, 8, 8, 6, 6, 9, 7, 6, 8, 6, 8, 7, 6, 8, 9, 7, 7, 9, 5, 7, 9, 9, 7, 7, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS The only fixed points for a(n) are 3, 4, 5, 6, 7, 10, 13, 22. If n > 25 and n != 2 (mod 3) then a(n) <= (n+9)/4. If n > 5 and n = 2 (mod 3) then a(n) <= (n+4)/3. [corrected by Jukka Kohonen, Feb 16 2022] It is conjectured that a(n) = o(log(n)). a(mn) <= a(m)+a(n)-1, by joining two graphs with m and n spanning trees at a single common vertex. - Jukka Kohonen, Feb 17 2022 LINKS Jukka Kohonen, Table of n, a(n) for n = 3..10000 Jernej Azarija, MathOverflow: Minimal graphs with a prescribed number of spanning trees Jernej Azarija and Riste Škrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, Mathematica Bohemica, Vol. 138 (2013), No. 2, 121--131. Dick Lipton, The Inverse Spanning Tree Problem Ladislav Nebeský, On the minimum number of vertices and edges in a graph with a given number of spanning trees, Časopis pro pěstování matematiky 98 (1973), 95-97. J. Sedláček, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517. EXAMPLE a(100000000) = 10 since K_10 has 100000000 spanning trees. From Jukka Kohonen, Feb 17 2022: (Start) a(47) = 11 since the following graph has 47 spanning trees: o-o-o-o / \ o--o---o--o \ / o--o--o (End) CROSSREFS Sequence in context: A111608 A126800 A245689 * A067628 A168093 A095254 Adjacent sequences: A182255 A182256 A182257 * A182259 A182260 A182261 KEYWORD nonn AUTHOR Jernej Azarija, Apr 21 2012 EXTENSIONS a(14)-a(46) from Jukka Kohonen, Feb 16 2022 a(47)-a(81) from Jukka Kohonen, Feb 17 2022 STATUS approved

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Last modified December 8 12:07 EST 2023. Contains 367678 sequences. (Running on oeis4.)