A085239 -id:A085239 - cp's OEIS frontend

A085242 a(n) = A085239(n+1) - 2.

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

Cf. A085239.

upto(L) = my(v2=2, v3=1, r=List()); while(v3Ruud H.G. van Tol, May 10 2024

from sympy import integer_log
def A085242(n): return int(6**integer_log(m:=3**n,6)[0]<<1>=m) # Chai Wah Wu, Feb 05 2025

Name corrected by Ruud H.G. van Tol, May 10 2024

A085241 a(n) = A085239(n+1) - A085239(n).

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1

Offset: 1

Reinhard Zumkeller, Jun 22 2003

sign

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

Original entry on oeis.org

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

Simon Plouffe and N. J. A. Sloane

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

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).

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.

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.

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

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

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

is(n)=n>>valuation(n,2)==1 || n==3^valuation(n,3) \\ Charles R Greathouse IV, Aug 29 2016

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

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

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

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

More terms from Reinhard Zumkeller, Jun 22 2003

A085238 Sort the numbers 2^i and 3^j. Then a(n) is the exponent of the n-th term.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 3, 5, 6, 4, 7, 5, 8, 9, 6, 10, 11, 7, 12, 8, 13, 14, 9, 15, 10, 16, 17, 11, 18, 19, 12, 20, 13, 21, 22, 14, 23, 15, 24, 25, 16, 26, 17, 27, 28, 18, 29, 30, 19, 31, 20, 32, 33, 21, 34, 22, 35, 36, 23, 37, 38, 24, 39, 25, 40, 41, 26, 42, 27, 43, 44, 28

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from T. D. Noe)

Cf. A006899, A085239, A085240.

a085238 n = e (mod x 2 + 2) x where
   x = a006899 n
   e b p = if p == 1 then 0 else 1 + e b (p `div` b)
-- Reinhard Zumkeller, Oct 09 2013

seq[lim_] := Module[{r2 = Range[0, Floor[Log2[lim]]], r3 = Range[0, Floor[Log[3, lim]]]}, Rest@ SortBy[Join[{#, 2^#} & /@ r2, {#, 3^#} & /@ r3], Last][[;; , 1]]]; seq[10^14] (* Amiram Eldar, Mar 25 2025 *)

do(lim)=my(v=List(vector(logint(lim\=1,2),i,1<my(t=valuation(n,2)); if(t, t, valuation(n,3)), Set(v)) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import integer_log
def A085238(n): return k+1 if 6**(k:=integer_log(m:=3**(n-1),6)[0])<<1Chai Wah Wu, Feb 04 2025

A006899(n) = A085239(n)^a(n).

a(A085240(n)) = a(n).

A006895 Parenthesized one way gives the powers of 2: (1), (2), (1+3), ..., another way the powers of 3: (1), (2+1), (3+6), ....

Original entry on oeis.org

1, 2, 1, 3, 6, 2, 16, 9, 23, 58, 6, 128, 109, 147, 512, 70, 954, 1233, 815, 4096, 1650, 6542, 13141, 3243, 32768, 23038, 42498, 131072, 3577, 258567, 272874, 251414, 1048576, 294333, 1802819, 2980150, 1214154, 8388608, 4746145, 12031071, 31015650, 2538782

Offset: 0

N. J. A. Sloane, K. S. Brown [ kevin2003(AT)delphi.com ]

Powers of 2 need 1 term or 2 terms parenthesized, whereas powers of 3 need 2 or 3 terms parenthesized, when 3 then the middle term is a power of 2. See A227928. - Reinhard Zumkeller, Oct 09 2013

			.  a(0)                                              =  _^0
.  a(1)                                              =  2^1
.  a(1) + a(2)            =    2 +   1               =  3^1
.  a(2) + a(3)            =    1 +   3        =    4 =  2^2
.  a(3) + a(4)            =    3 +   6        =    9 =  3^2
.  a(4) + a(5)            =    6 +   2        =    8 =  2^3
.  a(6)                                       =   16 =  2^4
.  a(5) + a(6) + a(7)     =    2 +  16 +   9  =   27 =  3^3
.  a(7) + a(8)            =    9 +  23        =   32 =  2^5
.  a(8) + a(9)            =   23 +  58        =   81 =  3^4
.  a(9) + a(10)           =   58 +   6        =   64 =  2^6
.  a(11)                                      =  128 =  2^7
.  a(10) + a(11) + a(12)  =    6 + 128 + 109  =  243 =  3^5
.  a(12) + a(13)          =  109 + 147        =  256 =  2^8
.  a(14)                                      =  512 =  2^9
.  a(13) + a(14) + a(15)  =  147 + 512 +  70  =  3^6 =  729 .

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000079, A000244, A006899, A108906, A085239.

a006895 n = a006895_list !! n
a006895_list = 1 : f 0 0 (tail a000079_list) (tail a000244_list) where
   f x y us'@(u:us) vs'@(v:vs)
     | x > 0     = (u - x) : f 0 (u - x + y) us vs'
     | y > v - u = (v - y) : f (v + x - y) 0 us' vs
     | otherwise =       u : f 0 (u + y) us vs'
-- Reinhard Zumkeller, Oct 09 2013

cp's OEIS Frontend

A085242 a(n) = A085239(n+1) - 2.

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A085241 a(n) = A085239(n+1) - A085239(n).

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

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A085238 Sort the numbers 2^i and 3^j. Then a(n) is the exponent of the n-th term.

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Programs

Haskell

Mathematica

PARI

Python

Formula

A006895 Parenthesized one way gives the powers of 2: (1), (2), (1+3), ..., another way the powers of 3: (1), (2+1), (3+6), ....

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Haskell