OFFSET
1,2
COMMENTS
This sequence is easily confused with A033845, which gives numbers of the form 2^i*3^j with i, j >= 1. Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024
These numbers were once called "harmonic numbers", see Lenstra links. - N. J. A. Sloane, Jul 03 2015
Successive numbers k such that phi(6k) = 2k. - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A088468: A160519(n) = A088468(a(n)). - Reinhard Zumkeller, May 16 2009
Also numbers that are divisible by neither 6k - 1 nor 6k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010
Also numbers m such that the rooted tree with Matula-Goebel number m has m antichains. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. An antichain is a nonempty set of mutually incomparable vertices. Example: m=4 is in the sequence because the corresponding rooted tree is \/=ARB (R is the root) having 4 antichains (A, R, B, AB). - Emeric Deutsch, Jan 30 2012
A204455(3*a(n)) = 3, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
The number of terms less than or equal to n is Sum_{i=0..floor(log_2(n))} floor(log_3(n/2^i) + 1), or Sum_{i=0..floor(log_3(n))} floor(log_2(n/3^i) + 1), which requires fewer terms to compute. - Robert G. Wilson v, Aug 17 2012
Named 3-friables in French. - Michel Marcus, Jul 17 2013
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
Range of values of A000005(n) (and also A181819(n)) for cubefree numbers n. - Matthew Vandermast, May 14 2014
A036561 is a permutation of this sequence. - L. Edson Jeffery, Sep 22 2014
The sum of the reciprocals of the 3-smooth numbers is equal to 3. Brief proof: 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + ... = (Sum_{k>=0} 1/2^k) * (Sum_{m>=0} 1/3^m) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3. - Bernard Schott, Feb 19 2019
Also those integers k for which, for every prime p > 3, p^(2k) - 1 == 0 (mod 24k). - Federico Provvedi, May 23 2022
For n>1, the exponents’ parity {parity(i), parity(j)} of one out of four consecutive terms is {odd, odd}. Therefore, for n>1, at least one out of every four consecutive terms is a Zumkeller number (A083207). If for the term whose parity is {even, odd}, even also means nonzero, then this term is also a Zumkeller number (as is the case with the last of the four consecutive terms 1296, 1458, 1536, 1728). - Ivan N. Ianakiev, Jul 10 2022
Except the initial terms 2, 3, 4, 8, 9 and 16, these are numbers k such that k^6 divides 6^k. Except the initial terms 2, 3, 4, 6, 8, 9, 16, 18 and 27, these are numbers k such that k^12 divides 12^k. - Mohammed Yaseen, Jul 21 2022
REFERENCES
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 654 pp. 85, 287-8, Ellipses Paris 2004.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.
R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
LINKS
Lei Zhou, Table of n, a(n) for n = 1..10000 (first 501 terms from Franklin T. Adams-Watters)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
Benoit Cloitre, a(n)/((1/sqrt(6))*exp(sqrt(2*log(2)*log(3)*n))) for 0<n<10^5
Natalia da Silva, Serban Raianu, and Hector Salgado, Differences of Harmonic Numbers and the abc-Conjecture, arXiv:1708.00620 [math.NT], 2017.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 252. Book's website
H. W. Lenstra Jr., Harmonic Numbers
H. W. Lenstra, Jr., Harmonic Numbers and the ABC-conjecture, Abstract of talk, May 30, 2001 [Annotated scanned copy]
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
D. J. Mintz, 2,3 sequence as a binary mixture, Fib. Quarterly, Vol. 19, No 4, Oct 1981, pp. 351-360.
I. Peterson, Medieval Harmony
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Smooth Number
FORMULA
An asymptotic formula for a(n) is roughly a(n) ~ 1/sqrt(6)*exp(sqrt(2*log(2)*log(3)*n)). - Benoit Cloitre, Nov 20 2001
A061987(n) = a(n + 1) - a(n), a(A084791(n)) = A084789(n), a(A084791(n) + 1) = A084790(n). - Reinhard Zumkeller, Jun 03 2003
Union of powers of 2 and 3 with n such that psi(n) = 2*n, where psi(n) = n*Product_(1 + 1/p) over all prime factors p of n = A001615(n). - Lekraj Beedassy, Sep 07 2004; corrected by Franklin T. Adams-Watters, Mar 19 2009
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} moebius(6*n)*x^n/(1 - x^n). - Paul D. Hanna, Sep 18 2011
MAPLE
A003586 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do numtheory[factorset](a) minus {2, 3} ; if % = {} then return a; end if; end do: end if; end proc: # R. J. Mathar, Feb 28 2011
with(numtheory): for i from 1 to 23328 do if(i/phi(i)=3)then print(i/6) fi od; # Gary Detlefs, Jun 28 2011
MATHEMATICA
a[1] = 1; j = 1; k = 1; n = 100; For[k = 2, k <= n, k++, If[2*a[k - j] < 3^j, a[k] = 2*a[k - j], {a[k] = 3^j, j++}]]; Table[a[i], {i, 1, n}] (* Hai He (hai(AT)mathteach.net) and Gilbert Traub, Dec 28 2004 *)
aa = {}; Do[If[EulerPhi[6 n] == 2 n, AppendTo[aa, n]], {n, 1, 1000}]; aa (* Artur Jasinski, Nov 05 2008 *)
fQ[n_] := Union[ MemberQ[{1, 5}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 6]] == {False}; fQ[1] = True; Select[ Range@ 4000, fQ] (* Robert G. Wilson v, Oct 26 2010 *)
powerOfTwo = 12; Select[Nest[Union@Join[#, 2*#, 3*#] &, {1}, powerOfTwo-1], # < 2^powerOfTwo &] (* Robert G. Wilson v and T. D. Noe, Mar 03 2011 *)
fQ[n_] := n == 3 EulerPhi@ n; Select[6 Range@ 4000, fQ]/6 (* Robert G. Wilson v, Jul 08 2011 *)
mx = 4000; Sort@ Flatten@ Table[2^i*3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
f[n_] := Block[{p2, p3 = 3^Range[0, Floor@ Log[3, n] + 1]}, p2 = 2^Floor[Log[2, n/p3] + 1]; Min[ Select[ p2*p3, IntegerQ]]]; NestList[f, 1, 54] (* Robert G. Wilson v, Aug 22 2012 *)
Select[Range@4000, Last@Map[First, FactorInteger@#] <= 3 &] (* Vincenzo Librandi, Aug 25 2016 *)
Select[Range[4000], Max[FactorInteger[#][[All, 1]]]<4&] (* Harvey P. Dale, Jan 11 2017 *)
PROG
(PARI) test(n)=for(p=2, 3, while(n%p==0, n/=p)); n==1;
for(n=1, 4000, if(test(n), print1(n", ")))
(PARI) list(lim)=my(v=List(), N); for(n=0, log(lim\1+.5)\log(3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011
(PARI) is_A003586(n)=n<5||vecmax(factor(n, 5)[, 1])<5 \\ M. F. Hasler, Jan 16 2015
(PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018
(Haskell)
import Data.Set (Set, singleton, insert, deleteFindMin)
smooth :: Set Integer -> [Integer]
smooth s = x : smooth (insert (3*x) $ insert (2*x) s')
where (x, s') = deleteFindMin s
a003586_list = smooth (singleton 1)
a003586 n = a003586_list !! (n-1)
-- Reinhard Zumkeller, Dec 16 2010
(Sage)
def isA003586(n) :
return not any(d != 2 and d != 3 for d in prime_divisors(n))
@CachedFunction
def A003586(n) :
if n == 1 : return 1
k = A003586(n-1) + 1
while not isA003586(k) : k += 1
return k
[A003586(n) for n in (1..55)] # Peter Luschny, Jul 20 2012
(Python)
from itertools import count, takewhile
def aupto(lim):
pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(0))))
pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(0))))
return sorted(c*d for c in pows2 for d in pows3 if c*d <= lim)
print(aupto(10**4)) # Michael S. Branicky, Jul 08 2022
(Python)
from sympy import integer_log
def A003586(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x, 3)[0]+1))
return bisection(f, n, n) # Chai Wah Wu, Sep 15 2024
(Python) # faster for initial segment of sequence
import heapq
from itertools import islice
def A003586gen(): # generator of terms
v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
while True:
v = heapq.heappop(h)
if v != oldv:
yield v
oldv = v
for p in psmooth_primes:
heapq.heappush(h, v*p)
print(list(islice(A003586gen(), 65))) # Michael S. Branicky, Sep 17 2024
(Magma) [n: n in [1..4000] | PrimeDivisors(n) subset [2, 3]]; // Bruno Berselli, Sep 24 2012
CROSSREFS
Cf. A051037, A002473, A051038, A080197, A080681, A080682, A117221, A105420, A062051, A117222, A117220, A090184, A131096, A131097, A186711, A186712, A186771, A088468, A061987, A080683 (p-smooth numbers with other values of p), A025613 (a subsequence).
Cf. A191475 (successive values of i), A191476 (successive values of j), A022330 (indices of the pure terms 2^i), A022331 (indices of the pure terms 3^j). - N. J. A. Sloane, May 26 2024
KEYWORD
nonn,easy,nice
AUTHOR
Paul Zimmermann, Dec 11 1996
EXTENSIONS
STATUS
approved