A006899 Numbers of the form 2^i or 3^j.

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

Offset: 1

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Complement of A033845 with respect to A003586. - 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

Deficient 3-smooth numbers, i.e., intersection of A005100 and A003586. - Amiram Eldar, Jun 03 2022

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 78.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 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.

Crossrefs

Union of A000079 and A000244. - Reinhard Zumkeller, Sep 25 2008

Cf. A085238, A085239.

Cf. A003586, A005100, A006530, A053669.

Cf. A170803, A235365, A235366, A236210.

A186927 and A186928 are subsequences.

Cf. A108906 (first differences), A006895, A227928.

Programs

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

Maple

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

Mathematica

seqMax = 10^20; Union[2^Range[0, Floor[Log[2, seqMax]]], 3^Range[0, Floor[Log[3, seqMax]]]] (* Stefan Steinerberger, Apr 08 2006 *)

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

PARI

a(n) = my(i0= logint(3^(n-1),6), i= logint(3^n,6)); if(i > i0, 2^i, my(j=logint(2^n,6)); 3^j) \\ Ruud H.G. van Tol, Nov 10 2022

Python

from sympy import integer_log
def A006899(n): return 1<Chai Wah Wu, Oct 01 2024

Formula

a(n) = A085239(n)^A085238(n). - Reinhard Zumkeller, Jun 22 2003

A086411(a(n)) = A086410(a(n)). - Reinhard Zumkeller, Sep 25 2008

A053669(a(n)) - A006530(a(n)) = (-1)^a(n) n > 1. - Anthony Browne, Jun 26 2016

Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Jun 03 2022

a(n)^(1/n) tends to 3^(log(2)/log(6)) = 2^(log(3)/log(6)) = 1.529592328491883538... - Vaclav Kotesovec, Sep 19 2024

Extensions

More terms from Reinhard Zumkeller, Jun 22 2003