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A246532 Smallest Meertens number in base n, or -1 if none exists. 2

%I

%S 2,10,200,6,54,100,216,4199040,81312000,-1,-1,-1,47250,-1,18,36

%N Smallest Meertens number in base n, or -1 if none exists.

%C A Meertens number in base n is a fixed point of the base n Godel encoding.

%C The base n Godel encoding of x is 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the base n representation of x.

%C The -1 entries are all conjectures.

%C In a computer search that included all numbers < 10^29 and bases <= 16, the only additional Meertens numbers found were 6 (base 2), 10 (base 2), 49000 (base 5), and 181400 (base 5).

%C There is no base 11 Meertens number < 11^44 ~= 6.6*10^45.

%C There is no base 12 Meertens number < 12^40 ~= 1.4*10^43.

%C There is no base 13 Meertens number < 13^39 ~= 2.7*10^43.

%C There is no base 15 Meertens number < 15^37 ~= 3.2*10^43.

%C Other terms: a(17) = 36, a(19) = 96, a(32) = 256, a(51) = 54. - _Chai Wah Wu_, Aug 28 2014

%C From _Chai Wah Wu_, Jul 20 2020: (Start)

%C All terms are even.

%C If n > 2 and a(n) != -1, then a(n) > n.

%C a(2*3^m-m) = 2*3^m for all m >= 0, i.e. a(n) > 0 for an infinite number of values of n.

%C Other terms: a(64) = a(4096) = 65536, a(71) = 216, a(160) = 324, a(323) = 1296, a(1455) = 2916, a(1942) = 5832, a(7775) = 46656, a(8294) = 82944, a(13118) = 26244.

%C (End)

%H David Applegate, <a href="/A246532/a246532.cc.txt">C++ program used to search for Meertens numbers</a>

%H Richard S. Bird, <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=44141&amp;fulltextType=RL&amp;fileId=S0956796897002931">Functional Pearl: Meertens number</a>, Journal of Functional Programming 8 (1), Jan 1998, 83-88.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Meertens_number">Meertens number</a>

%H Chai Wah Wu, <a href="http://domino.watson.ibm.com/library/CyberDig.nsf/papers/F24675F6E614358F85257E38004D39F0/$File/rc25531.pdf">Meertens Number and Its Variations</a>, IBM Research Report RC25531 (WAT1504-032) April 2015.

%H Chai Wah Wu, <a href="http://arxiv.org/abs/1603.08493">Meertens Number and Its Variations</a>, arXiv:1603.08493 [math.NT], 2016.

%e 100 is a base 7 Meertens number because 100 = 202_7 = 2^2 * 3^0 * 5^2.

%e 4199040 is a base 9 Meertens number because 4199040 = 7810000_9 = 2^7 * 3^8 * 5^1.

%Y Cf. A189398 (base 10 Godel encoding), A110765 (base 2 Godel encoding).

%K sign,more

%O 2,1

%A _David Applegate_, Aug 28 2014

%E a(17) from _Chai Wah Wu_, Jul 19 2020

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Last modified May 7 18:13 EDT 2021. Contains 343652 sequences. (Running on oeis4.)