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A006899
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Numbers of the form 2^i or 3^j.
(Formerly M0588)
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29
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1, 2, 3, 4, 8, 9, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 14348907, 16777216, 33554432
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OFFSET
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1,2
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COMMENTS
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Complement of A033845 with respect to A003586; A086411(a(n)) = A086410(a(n)). - Reinhard Zumkeller, Sep 25 2008
In the 14th century, Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1, 2), (2, 3), (3, 4), and (8, 9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014
Numbers n such that absolute value of the greatest prime factor of n minus the smallest prime not dividing n is 1 (that is, abs(A006530(n)-A053669(n)) = 1). - Anthony Browne, Jun 26 2016
1 and numbers k such that k = phi(k) + phi(2*k)/2. - Paolo P. Lava, Oct 26 2017
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REFERENCES
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G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..500
Boris Alexeev, Minimal DFAs for testing divisibility, arXiv:cs/0309052 [cs.CC], 2003.
Jung-Chao Ban, Wen-Guei Hu, and Song-Sun Lin, Pattern generation problems arising in multiplicative integer systems, arXiv preprint arXiv:1207.7154 [math.DS], 2012.
Lukas Spiegelhofer, Collisions of the binary and ternary sum-of-digits functions, arXiv:2105.11173 [math.NT], 2021.
Eric Weisstein's World of Mathematics, Pillai's Theorem.
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FORMULA
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a(n) = A085239(n)^A085238(n). - Reinhard Zumkeller, Jun 22 2003
A053669(a(n)) - A006530(a(n)) = (-1)^a(n) n > 1. - Anthony Browne, Jun 26 2016
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MAPLE
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A:={seq(2^n, n=0..63)}: B:={seq(3^n, n=0..40)}: C:=sort(convert(A union B, list)): seq(C[j], j=1..39); # Emeric Deutsch, Aug 03 2005
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MATHEMATICA
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seqMax = 10^20; Union[2^Range[0, Floor[Log[2, seqMax]]], 3^Range[0, Floor[Log[3, seqMax]]]] (* Stefan Steinerberger, Apr 08 2006 *)
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PROG
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(Haskell)
a006899 n = a006899_list !! (n-1)
a006899_list = 1 : m (tail a000079_list) (tail a000244_list) where
m us'@(u:us) vs'@(v:vs) = if u < v then u : m us vs' else v : m us' vs
-- Reinhard Zumkeller, Oct 09 2013
(PARI) is(n)=n>>valuation(n, 2)==1 || n==3^valuation(n, 3) \\ Charles R Greathouse IV, Aug 29 2016
(PARI) upto(n) = my(res = vector(logint(n, 2) + logint(n, 3) + 1), t = 1); res[1] = 1; for(i = 2, 3, for(j = 1, logint(n, i), t++; res[t] = i^j)); vecsort(res) \\ David A. Corneth, Oct 26 2017
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CROSSREFS
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Union of A000079 and A000244. - Reinhard Zumkeller, Sep 25 2008
Cf. A085238, A085239.
Cf. A006530, A053669.
Cf. A170803, A235365, A235366, A236210.
A186927 and A186928 are subsequences.
Cf. A108906 (first differences), A006895, A227928.
Sequence in context: A068317 A074311 A076382 * A256179 A078830 A304521
Adjacent sequences: A006896 A006897 A006898 * A006900 A006901 A006902
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Simon Plouffe and N. J. A. Sloane
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EXTENSIONS
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More terms from Reinhard Zumkeller, Jun 22 2003
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STATUS
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approved
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