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%I #12 Jun 05 2015 17:23:20
%S 1,5,103,4985,641687,326552783
%N The smallest number without double base representation of length n.
%C Portion of abstract from Dimitrov/Howe reference: A double-base representation of an integer n is an expression n = n_1 + ... + n_r, where the n_i are (positive or negative) integers that are divisible by no primes other than 2 or 3; the length of the representation is the number r of terms. It is known that there is a constant a > 0 such that every integer n has a double-base representation of length at most a log n / log log n.
%H Vassil S. Dimitrov and Everett W. Howe, <a href="http://www.ams.org/journals/proc/2011-139-10/S0002-9939-2011-10764-0/home.html">Lower bounds on the lengths of double-base representations</a>, Proc. Amer. Math. Soc. 139 (2011), 3423-3430. Also <a href="http://arxiv.org/abs/1001.4133">arXiv:1001.4133</a>.
%H Vassil S. Dimitrov and Everett W. Howe, <a href="http://alumnus.caltech.edu/~however/papers/paper34.html">Magma and C programs verifying these entries</a>
%H Vassil Dimitrov, Laurent Imbert, and Pradeep Kumar Mishra, <a href="http://dx.doi.org/10.1007/11593447_4">Efficient and secure elliptic curve point multiplication using double-base chains</a>, Advances in cryptology, ASIACRYPT 2005, Lecture Notes in Comput. Sci., vol. 3788, Springer, Berlin, 2005, pp. 59-78.
%H Vassil Dimitrov, Laurent Imbert, and Pradeep K. Mishra, <a href="http://dx.doi.org/10.1090/S0025-5718-07-02048-0">The double-base number system and its application to elliptic curve cryptography</a>, Math. Comp. 77 (2008), no. 262, 1075-1104.
%H Pradeep Kumar Mishra and Vassil Dimitrov, <a href="http://dx.doi.org/10.3934/amc.2008.2.159">A combinatorial interpretation of double base number system and some consequences</a>, Adv. Math. Commun. 2 (2008), no. 2, 159-173.
%e For example, 103 can be written in many ways as the sum of 3 integers, each with no prime divisors other than 2 and 3 (e.g. 103 = (-1) + (-4) + 108), but it cannot be written as the sum of 2 such integers. 103 is the smallest positive integer that requires more than 2 terms.
%Y Cf. A003586, A018899.
%K hard,nonn
%O 0,2
%A _Jonathan Vos Post_, Jan 26 2010
%E Prepended initial terms, updated references, modified comment, added example by _Everett W. Howe_, Jun 05 2015