A018899 Smallest positive integer not representable as the sum of at most n distinct numbers of form 2^a*3^b.
1, 5, 23, 431, 18431, 3448733, 1441896119
Offset: 0
Examples
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.
Links
- Valérie Berthé and Laurent Imbert, Diophantine Approximation, Ostrowski Numeration and the Double-Base Number System, Discrete Mathematics and Theoretical Computer Science, vol. 11 (2009) no 1, pp. 153-172.
- Valérie Berthé's Publications.
- Daniel Krenn, Vorapong Suppakitpaisarn, and Stephan Wagner, Multi-Base Representations and their Minimal Hamming Weight, Numeration 2018 (2018): 24.
- Mike Oakes, Posting to primenumbers list, Aug 30 2003.
- Mark Underwood and others, The "twothree" numbers, digest of 10 messages in primenumbers Yahoo group, Aug 29 - Aug 30, 2003.
Programs
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Mathematica
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 *)
Extensions
a(6) = 1441896119 found by Mike Oakes, Sep 07 2003