A006899 -id:A006899 - cp's OEIS frontend

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

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

A097275 Least integer "mod 2 prime signatures" k ordered by number of primitive Pythagorean triples with leg = k.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 18, 8, 15, 60, 30, 9, 24, 105, 420, 54, 16, 36, 120, 840, 4620, 90, 27, 45, 180, 1155, 9240, 60060, 162, 32, 48, 240, 1260, 13860, 120120, 1021020, 210, 64, 72, 315, 1680, 15015, 180180, 2042040, 19399380, 270, 81, 96, 360, 2520, 18480, 240240

Offset: 0

Ray Chandler, Aug 22 2004

Row 0 of table represents "mod 2 prime signature" values k such that no PPTs have leg=k.
Row n of table, n>0, represents "mod 2 prime signature" values k such that 2^(n-1) PPTs have leg=k. Table read by antidiagonals.
For n=2^a_0*p_1^a_1*...*p_n^a_n where p_i is odd prime and a_1>=a_2>=...>=a_n, define "mod 2 prime signature" to be ordered prime exponents (a_0,a_1,...,a_n).
Least integer with a given "mod 2 prime signature" is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.

			Table begins:
0: 1,2,6,18,30,54,90,162,210,270,...
1: 3,4,8,9,16,27,32,64,81,128,...
2: 12,15,24,36,45,48,72,96,108,135,...
4: 60,105,120,180,240,315,360,480,540,720,...
8: 420,840,1155,1260,1680,2520,3360,3465,3780,5040,...
16: 4620,9240,13860,15015,18480,27720,36960,41580,45045,55440,...
32: 60060,120120,180180,240240,255255,360360,480480,540540,...
64: 1021020,2042040,3063060,4084080,4849845,6126120,8168160,...
128: 19399380,38798760,58198140,77597520,111546435,116396280,...
256: 446185740,892371480,1338557220,1784742960,2677114440,...

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A097272, A097273, A097274.

Row 1 is A006899 except for starting point.

A227928 Powers of 2 or of 3 in order as occurring in the two ways of parenthesizing the terms in A006895.

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 16, 27, 32, 81, 64, 128, 243, 256, 512, 729, 1024, 2187, 2048, 4096, 6561, 8192, 19683, 16384, 32768, 59049, 65536, 131072, 177147, 262144, 531441, 524288, 1048576, 1594323, 2097152, 4782969, 4194304, 8388608, 14348907, 16777216, 43046721

Offset: 1

Reinhard Zumkeller, Oct 09 2013

nonn

Permutation of A006899, a(n) = A006899(n+k) for some k in {-1,0,1}.

			.  n  |  a(n)                       let x denote A006895
. ----+------
.   1 |    1                     =  x(0)
.   2 |    2                     =  x(1)
.   3 |    3  =   2 +   1        =  x(1) + x(2)
.   4 |    4  =   1 +   3        =  x(2) + x(3)
.   5 |    9  =   3 +   6        =  x(3) + x(4)
.   6 |    8  =   6 +   2        =  x(4) + x(5)
.   7 |   16                     =  x(6)
.   8 |   27  =   2 +  16 +   9  =  x(5) + x(6) + x(7)
.   9 |   32  =   9 +  23        =  x(7) + x(8)
.  10 |   81  =  23 +  58        =  x(8) + x(9)
.  11 |   64  =  58 +   6        =  x(9) + x(10)
.  12 |  128                     =  x(11)
.  13 |  243  =   6 + 128 + 109  =  x(10) + x(11) + x(12)
.  14 |  256  = 109 + 147        =  x(12) + x(13)
.  15 |  512                     =  x(14)
.  16 |  729  = 147 + 512 +  70  =  x(13) + x(14) + x(15) .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

a227928 n = a227928_list !! (n-1)
a227928_list = 1 : f 0 0 (tail a000079_list) (tail a000244_list) where
   f x y us'@(u:us) vs'@(v:vs)
     | x > 0     = u : f 0 (u - x + y) us vs'
     | y > v - u = v : f (v + x - y) 0 us' vs
     | otherwise = u : f 0 (u + y) us vs'

A206807 Position of 3^n when {2^j} and {3^k} are jointly ranked; complement of A206805.

Original entry on oeis.org

2, 5, 7, 10, 12, 15, 18, 20, 23, 25, 28, 31, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 69, 72, 74, 77, 80, 82, 85, 87, 90, 93, 95, 98, 100, 103, 105, 108, 111, 113, 116, 118, 121, 124, 126, 129, 131, 134, 137, 139, 142, 144, 147, 149, 152, 155

Offset: 1

Clark Kimberling, Feb 16 2012

nonn

The joint ranking is for j >= 1 and k >= 1, so that the sets {2^j} and {3^k} are disjoint.

			The joint ranking begins with 2,3,4,8,9,16,27,32,64,81,128,243,256, so that
A206805 = (1,3,4,6,8,9,11,13,...)
A206807 = (2,5,7,10,12,...)

Cf. A006899, A056576, A098294, A206805, A122437.

f[n_] := 2^n; g[n_] := 3^n; z = 200;
c = Table[f[n], {n, 1, z}]; s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]]           (* A206805 *)
Table[n + Floor[n*Log[3, 2]], {n, 1, 50}] (* A206805 *)
Flatten[Table[q[n], {n, 1, z}]]           (* this sequence *)
Table[n + Floor[n*Log[2, 3]], {n, 1, 50}] (* this sequence as a table *)

a(n) = logint(3^n, 2) + n; \\ Ruud H.G. van Tol, Dec 10 2023

a(n) = n + floor(n*log_2(3)).
A206805(n) = n + floor(n*log_3(2)).
a(n) = n + A056576(n). - Michel Marcus, Dec 12 2023
a(n) = A098294(n) + 2*n - 1. - Ruud H.G. van Tol, Jan 22 2024

A372823 Sequence formed as follows: for each k >= 0, insert between 3^k and 3^(k+1) the least power of 2 that is in the interval [3^k, 3^(k+1)], and then arrange the resulting numbers in nondecreasing order.

Original entry on oeis.org

1, 1, 3, 4, 9, 16, 27, 32, 81, 128, 243, 256, 729, 1024, 2187, 4096, 6561, 8192, 19683, 32768, 59049, 65536, 177147, 262144, 531441, 1048576, 1594323, 2097152, 4782969, 8388608, 14348907, 16777216, 43046721, 67108864, 129140163, 134217728, 387420489

Offset: 0

Clark Kimberling, May 18 2024

			3^0 <= 2^0 < 3^1 < 2^2 < 3^2 < 2^4 < 3^3 < ...

Cf. A000079, A000244, A006899, A372824, A372825, A372826.

[seq(op([3^i, 2^ceil(log[2](3^i))]),i=0..50)]; # Robert Israel, May 22 2024

a[n_] := If[EvenQ[n], 3^(n/2), 2^Ceiling[((n - 1)/2) Log[3]/Log[2]]]
Table[a[n], {n, 0, 37}]

A372824 Sequence formed as follows: for each k >= 0, insert between 3^k and 3^(k+1) the greatest power of 2 that is in the interval [3^k, 3^(k+1)], and then arrange the resulting numbers in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 8, 9, 16, 27, 64, 81, 128, 243, 512, 729, 2048, 2187, 4096, 6561, 16384, 19683, 32768, 59049, 131072, 177147, 524288, 531441, 1048576, 1594323, 4194304, 4782969, 8388608, 14348907, 33554432, 43046721, 67108864, 129140163, 268435456, 387420489

Offset: 0

Clark Kimberling, May 18 2024

			3^0 <= 2^1 < 3^1 < 2^3 < 3^2 < 2^4 < 3^3 < ...

Cf. A000079, A000244, A006899, A372823, A372825, A372826.

Interspersion of A000244 and A098232.

a[n_] := If[EvenQ[n], 3^(n/2), 2^Floor[((n + 1)/2) Log[3]/Log[2]]]
Table[a[n], {n, 0, 37}]

a(n) = if (n%2, 3^(n\2), 2^logint(3^(n/2), 2)); \\ Michel Marcus, May 23 2024

A096067 Number of 3-smooth numbers between successive numbers that are powers of 2 or of 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 0, 3, 1, 2, 4, 0, 5, 2, 3, 6, 0, 6, 5, 2, 8, 2, 6, 8, 1, 10, 4, 6, 11, 0, 11, 8, 4, 13, 3, 10, 12, 2, 15, 6, 9, 16, 0, 17, 9, 8, 18, 2, 16, 14, 5, 20, 6, 14, 19, 2, 22, 10, 12, 23, 1, 22, 16, 8, 25, 6, 19, 22, 4, 27, 11, 16, 28, 0, 29, 16, 13, 30, 4

Offset: 1

Reinhard Zumkeller, Jul 21 2004

a(n) = {k: A006899(n) < A003586(k) < A006899(n+1)}.

			n=16: there are three 3-smooth numbers between A006899(16)=3^6=729 and A006899(17)=2^10=1024: A003586(38)=2^8*3=768, A003586(39)=2^5*3^3=864 and A003586(40)=2^2*3^5=972, therefore a(16)=3.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A003586, A006899, A071521.

spi[n_] := Sum[Floor@Log[2, n/3^k] + 1, {k, 0, Floor@Log[3, n]}];
seq[n_] := Module[{a = Table[0, {n}], p = 1, s = 1}, For[i = 1, i <= Length[a], i++, p = Min[2^(1 + Floor@Log[2, p]), 3^(1 + Floor@Log[3, p])]; With[{t = spi[p]}, a[[i]] = t - s - 1; s = t]]; a];
seq[100] (* Jean-François Alcover, Dec 17 2021, after Andrew Howroyd's PARI code *)

\\ here spi(n) is A071521(n).
spi(n)={sum(k=0, logint(n, 3), logint(n\3^k, 2)+1)}
seq(n)={my(a=vector(n), p=1, s=1); for(i=1, #a, p=min(2^(1+logint(p,2)), 3^(1+logint(p,3))); my(t=spi(p)); a[i]=t-s-1; s=t); a} \\ Andrew Howroyd, Jan 07 2020

Terms a(40) and beyond from Andrew Howroyd, Jan 06 2020

A096075 Least common multiple of first n 3-smooth numbers.

Original entry on oeis.org

1, 2, 6, 12, 12, 24, 72, 72, 144, 144, 144, 432, 864, 864, 864, 864, 1728, 1728, 5184, 5184, 5184, 10368, 10368, 10368, 10368, 10368, 31104, 62208, 62208, 62208, 62208, 62208, 62208, 124416, 124416, 124416, 373248, 373248, 373248, 373248, 746496

Offset: 1

Reinhard Zumkeller, Jul 21 2004

nonn

Subsequence of A003586.

			The first seven 3-smooth numbers are {1, 2, 3, 4, 6, 8, 9} and their lcm is 72. - _David A. Corneth_, Jul 13 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's.

Cf. A003418, A003586, A006899, A022330, A022331, A051451.

seq[max_] := Module[{sm3 = Sort[Flatten[Table[2^i*3^j, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]], e2, e3}, e2 = FoldList[Max, IntegerExponent[sm3, 2]]; e3 = FoldList[Max, IntegerExponent[sm3, 3]]; 2^e2*3^e3]; seq[1000] (* Amiram Eldar, Jul 13 2023 *)

a(n) > a(n-1) iff A003586(n) is a power of 2 or of 3 (cf. A006899, A022330, A022331).

A258439 Powers of 3 alternating with powers of 2.

Original entry on oeis.org

1, 1, 3, 2, 9, 4, 27, 8, 81, 16, 243, 32, 729, 64, 2187, 128, 6561, 256, 19683, 512, 59049, 1024, 177147, 2048, 531441, 4096, 1594323, 8192, 4782969, 16384, 14348907, 32768, 43046721, 65536, 129140163, 131072, 387420489, 262144, 1162261467

Offset: 0

Luce ETIENNE, May 30 2015

a(n)*A098293(n) = A000400(floor(n/2)).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-6).

Cf. A098293, A006899.

Flat(List([0..20],n->[3^n,2^n])); # Muniru A Asiru, Jul 16 2018

&cat[[3^n, 2^n]: n in [0..35]]; // Vincenzo Librandi, Jul 17 2018

seq(op([3^n,2^n]),n=0..20); # Muniru A Asiru, Jul 16 2018

Flatten[Table[{3^n, 2^n}, {n, 0, 25}]] (* Vincenzo Librandi, Jul 17 2018 *)

Vec(-(3*x^3+2*x^2-x-1)/((2*x^2-1)*(3*x^2-1)) + O(x^100)) \\ Colin Barker, May 30 2015

a(n) = ((5+(-1)^n)/2)^((2*n-1+(-1)^n)/4).
a(n) = 5*a(n-2)-6*a(n-4). - Colin Barker, May 30 2015
G.f.: -(3*x^3+2*x^2-x-1) / ((2*x^2-1)*(3*x^2-1)). - Colin Barker, May 30 2015

A328415 Numbers k such that (Z/mZ)* = C_2 X C_(2k) has exactly one solution, where (Z/mZ)* is the multiplicative group of integers modulo m.

Original entry on oeis.org

4, 16, 27, 32, 64, 256, 512, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 14348907, 16777216, 33554432, 67108864, 134217728, 268435456, 387420489, 536870912, 1073741824

Offset: 1

Jianing Song, Oct 14 2019

nonn

Numbers k being powers of 2 or 3 such that 2*k+1 is not prime.
Proof. If m is a solution to (Z/mZ)* = C_2 X C_(2k) such that m is odd, then 2*m is also a solution, and vice versa. So if there is only one solution to (Z/mZ)* = C_2 X C_(2k), m must be a multiple of 4. If 8 divides m and m has odd prime factors, or if m has at least two distinct odd prime factors, then A046072(m) >= 3, a contradiction. So m = 2^e, e >= 3 or m = 4*p^e, p odd prime and e >= 1. If m = 4*p^e and p >= 5, then (Z/(3*p^e)Z)* = (Z/mZ)*. So we have m = 2^e, e >= 3 or m = 4*3^e, e >= 1, then (Z/mZ)* = C_2 X C_(2*2^(e-3)) or (Z/mZ)* = C_2 X C_(2*3^(e-1)).
If k = 2^(e-3) > 1 and p = 2*k+1 is prime, then (Z/(3*p)Z)* = (Z/(2^e)Z)*; if k = 3^(e-1) > 1 and p = 2*k+1 is prime, then (Z/(3*p)Z)* = (Z/(4*3^e)Z)*; on the other hand, if k is a power of 2 or a power of 3 such that 2*k+1 is not prime, then (Z/mZ)* = C_2 X C_(2k) indeed has only one solution.

			The only solution to (Z/mZ)* = C_2 X C_54 is m = 324, so 54/2 = 27 is a term.

Wikipedia, Multiplicative group of integers modulo n

Cf. A328412.

select(i->!isprime(2*i+1), upto(10^9)) \\ See A006899 for the function upto(n)

cp's OEIS Frontend

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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A097275 Least integer "mod 2 prime signatures" k ordered by number of primitive Pythagorean triples with leg = k.

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A227928 Powers of 2 or of 3 in order as occurring in the two ways of parenthesizing the terms in A006895.

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Haskell

A206807 Position of 3^n when {2^j} and {3^k} are jointly ranked; complement of A206805.

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A372823 Sequence formed as follows: for each k >= 0, insert between 3^k and 3^(k+1) the least power of 2 that is in the interval [3^k, 3^(k+1)], and then arrange the resulting numbers in nondecreasing order.

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Mathematica

A372824 Sequence formed as follows: for each k >= 0, insert between 3^k and 3^(k+1) the greatest power of 2 that is in the interval [3^k, 3^(k+1)], and then arrange the resulting numbers in nondecreasing order.

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Mathematica

PARI

A096067 Number of 3-smooth numbers between successive numbers that are powers of 2 or of 3.

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Mathematica

PARI

Extensions

A096075 Least common multiple of first n 3-smooth numbers.

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