login
Smallest positive integer not representable as the sum of at most n distinct numbers of form 2^a*3^b.
5

%I #48 Jan 29 2024 01:54:26

%S 1,5,23,431,18431,3448733,1441896119

%N Smallest positive integer not representable as the sum of at most n distinct numbers of form 2^a*3^b.

%H Valérie Berthé and Laurent Imbert, <a href="https://doi.org/10.46298/dmtcs.450">Diophantine Approximation, Ostrowski Numeration and the Double-Base Number System</a>, Discrete Mathematics and Theoretical Computer Science, vol. 11 (2009) no 1, pp. 153-172.

%H Valérie Berthé's <a href="https://www.irif.fr/~berthe/publi.html">Publications</a>.

%H Daniel Krenn, Vorapong Suppakitpaisarn, and Stephan Wagner, <a href="https://numeration2018.sciencesconf.org/data/pages/krenn.pdf">Multi-Base Representations and their Minimal Hamming Weight</a>, Numeration 2018 (2018): 24.

%H Mike Oakes, <a href="http://groups.yahoo.com/group/primenumbers/message/13418">Posting to primenumbers list, Aug 30 2003</a>.

%H Mark Underwood and others, <a href="/A018899/a018899.txt">The "twothree" numbers</a>, digest of 10 messages in primenumbers Yahoo group, Aug 29 - Aug 30, 2003.

%e a(2) = 23 because all smaller positive integers can be expressed as the sum of at most two numbers of the form 2^a * 3^b but not 23: 1=1, 2=2, 3=1+2, 4=1+3, 5=2+3, 6=2+4, 7=3+4, 8=2+6, 9=3+6, 10=4+6, 11=3+8, 12=4+8, 13=4+9, 14=6+8, 15=6+9, 16=4+12, 17=8+9, 18=6+12, 19=3+16, 20=8+12, 21=9+12, 22=6+16.

%t With[{s = Sort@ Flatten@ Table[2^a * 3^b, {a, 0, Log2@ #}, {b, 0, Log[3, #/(2^a)]}] &[2^16]}, {1}~Join~Array[Block[{k = 1}, While[! FreeQ[#, k], k++]; k] &@ Union[Total /@ Permutations[s, #]] &, 3]] (* _Michael De Vlieger_, Sep 04 2018 *)

%Y Cf. A003586, A172116, A237442.

%K nonn,more

%O 0,2

%A Vassil Dimitrov (vassil(AT)Engn.Uwindsor.Ca), Aug 15 1996

%E a(6) = 1441896119 found by _Mike Oakes_, Sep 07 2003