A235366 -id:A235366 - cp's OEIS frontend

A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3456, 3888

Offset: 1

Paul Zimmermann, Dec 11 1996

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
Also the sorted union of A000244 and A007694. - Lei Zhou, Apr 19 2017
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
In music theory, a comma is a ratio, close to 1 (typically less than 1.04), between two natural numbers divisible by only small primes (typically single digit). In this sequence, a(131) / a(130) = 531441 / 524288 ~ 1.013643 is the Pythagorean comma (A221363), the difference between 12 perfect fifths and 7 octaves. - Hal M. Switkay, Mar 23 2025

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.

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

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

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. also A235365, A235366, A236210, A036561.
Cf. also A000244, A007694. - Lei Zhou, Apr 19 2017
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
Cf. A221363.

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

[n: n in [1..4000] | PrimeDivisors(n) subset [2,3]]; // Bruno Berselli, Sep 24 2012

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

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 *)

test(n)=for(p=2,3, while(n%p==0, n/=p)); n==1;
for(n=1,4000,if(test(n),print1(n",")))

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

is_A003586(n)=n<5||vecmax(factor(n,5)[, 1])<5 \\ M. F. Hasler, Jan 16 2015

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

from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022

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

# 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
(C++) // Returns A003586 <= threshold without approximations nor sorting
#include 
std::forward_list A003586(const int threshold) {
    std::forward_list sequence;
    auto start_it = sequence.before_begin();
    for (int i = 1; i <= threshold; i *= 2) {
        for (int inc = 1; std::next(start_it) != sequence.end() && inc <= i; inc *= 3)
            ++start_it;
        auto it = start_it;
        for (int j = 1; i * j <= threshold; j *= 3) {
            sequence.emplace_after(it, i * j);
            for (int inc = 1; std::next(it) != sequence.end() && inc <= i; inc *= 2)
                ++it;
        }
    }
    return sequence;
} // Eben Gino Lester, Apr 17 2025

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

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
a(n) = 2^A022328(n)*3^A022329(n). - N. J. A. Sloane, 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
a(n) = A007694(n+1)/2. - Lei Zhou, Apr 19 2017

Deleted claim that this sequence is union of 2^n (A000079) and 3^n (A000244) sequences -- this does not include the terms which are not pure powers. - Walter Roscello (wroscello(AT)comcast.net), Nov 16 2008

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

A061987 Number of times n-th distinct value is repeated in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984; also number of times n-th distinct row is repeated in square array T(n,k) = T(n-1,k) + T(n-1,floor(k/2)) + T(n-1,floor(k/3)) with T(0,0) = 1, i.e., in A061980.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 6, 3, 5, 4, 12, 6, 10, 8, 9, 15, 12, 20, 16, 18, 30, 24, 27, 13, 32, 36, 60, 48, 54, 26, 64, 72, 81, 39, 96, 108, 52, 128, 144, 162, 78, 192, 216, 104, 139, 117, 288, 324, 156, 384, 432, 208, 278, 234, 576, 648, 312, 417, 351, 864, 416, 556

Offset: 0

Henry Bottomley, May 24 2001

nonn

For n > 0: a(n) = A003586(n+1) - A003586(n) and a(A084791(n)) = A084788(n).
Also number of times A160519(n+1) is repeated in A088468. - Reinhard Zumkeller, May 16 2009
In the 14th century Levi Ben Gerson proved that a(n) > 1 for all n > 6; see A003586, A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Smooth Number

import Data.List (group)
a061987 n = a061987_list !! n
a061987_list = map length $ group a061984_list
-- Reinhard Zumkeller, Jan 11 2014

a(n) = A061986(A061985(n)).

More terms from Reinhard Zumkeller, Jun 03 2003

A235365 Smallest odd prime factor of 3^n + 1, for n > 1.

Original entry on oeis.org

5, 7, 41, 61, 5, 547, 17, 7, 5, 67, 41, 398581, 5, 7, 21523361, 103, 5, 2851, 41, 7, 5, 23535794707, 17, 61, 5, 7, 41, 523, 5, 6883, 926510094425921, 7, 5, 61, 41, 18427, 5, 7, 17, 33703, 5, 82064241848634269407, 41, 7, 5, 16921, 76801, 547, 5, 7, 41, 78719947, 5, 61, 17, 7, 5, 3187, 41

Offset: 2

Jonathan Sondow, Jan 19 2014

nonn

Levi Ben Gerson (1288-1344) proved that 3^n + 1 = 2^m has no solution in integers if n > 1, by showing that 3^n + l has an odd prime factor. His proof uses remainders after division of powers of 3 by 8 and powers of 2 by 8; see the Lenstra and Peterson links. For an elegant short proof, see the Franklin link.

			3^2 + 1 = 10 = 2*5, so a(2) = 5.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.

Table of n, a(n) for n = 2..768
Philip Franklin, Problem 2927, Amer. Math. Monthly, 30 (1923), p. 81.
Aaron Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
Hendrik Lenstra Harmonic Numbers, MSRI, 1998.
J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
Ivars Peterson, Medieval Harmony, Math Trek, MAA, 2012.
Wikipedia, Gersonides

See A235366 for 3^n - 1.

Cf. also A003586 (products 2^m * 3^n), A006899, A061987, A108906.

[PrimeDivisors(3^n +1)[2]: n in [2..60] ] ; // Vincenzo Librandi, Mar 16 2019

Table[FactorInteger[3^n + 1][[2, 1]], {n, 2, 50}]

a(2+4n) = 5 as 3^(2+4n) + 1 = (3^2)*(3^4)^n + 1 = 9*81^n + 1 = 9*(80+1)^n + 1 == 9 + 1 == 0 (mod 5).

a(3+6n) = 7 as 3^(3+6n) + 1 = (3^3)*(3^6)^n + 1 = 27*729^n + 1 = 27*(728+1)^n + 1 == 27 + 1 == 0 (mod 7), but 27 * 729^n + 1 == 2*(-1)^n + 1 !== 0 (mod 5).

Terms to a(132) in b-file from Vincenzo Librandi, Mar 16 2019
a(133)-a(658) in b-file from Amiram Eldar, Feb 05 2020
a(659)-a(768) in b-file from Max Alekseyev, Apr 27 2022

A108906 First differences of A006899.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 11, 5, 32, 17, 47, 115, 13, 256, 217, 295, 1024, 139, 1909, 2465, 1631, 8192, 3299, 13085, 26281, 6487, 65536, 46075, 84997, 262144, 7153, 517135, 545747, 502829, 2097152, 588665, 3605639, 5960299, 2428309, 16777216, 9492289

Offset: 1

Ali A. Tanara (tanara(AT)khayam.ut.ac.ir), Jul 17 2005

In the 14th century Levi Ben Gerson proved that this sequence contains only four 1s; see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014

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

Cf. A006899.

Cf. also A235365, A235366, A236210.

a108906 n = a108906_list !! (n-1)
a108906_list = zipWith (-) (tail a006899_list) a006899_list
-- 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]-C[j-1],j=2..44); # Emeric Deutsch, Aug 03 2005

nn = 10^20; t = Union[ 2^Range[0, Floor[Log[2, nn]]], 3^Range[0, Floor[Log[3, nn]]]]; Differences@ t (* Robert G. Wilson v, May 26 2014 *)

from sympy import integer_log
def A108906(n):
    m, m2 = 3**(n-1), 1<Chai Wah Wu, Oct 01 2024

More terms from Emeric Deutsch, Aug 03 2005

A218356 Minimal order of degree-n irreducible polynomials over GF(3).

Original entry on oeis.org

1, 4, 13, 5, 11, 7, 1093, 32, 757, 44, 23, 35, 797161, 547, 143, 17, 1871, 19, 1597, 25, 14209, 67, 47, 224, 8951, 398581, 109, 29, 59, 31, 683, 128, 299, 103, 71, 95, 13097927, 2851, 169, 352, 83, 43, 431, 115, 181, 188, 1223, 97, 491, 151, 12853, 53, 107

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 3^n.

For n > 2, a(n) <= A143663(n). For odd prime n, a(n) = A143663(n). - Max Alekseyev, Apr 30 2022

Max Alekseyev, Table of n, a(n) for n = 1..796 (first 100 terms from Alois P. Heinz)

Cf. A143663, A212906, A213224, A235366.

M:= proc(n) M(n):= numtheory[divisors](3^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..60);

M[n_] := M[n] = Divisors[3^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(3^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212906(n,1) = A213224(n,2).

A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 8, 9

Offset: 1

Jonathan Sondow, Jan 20 2014

Philippe de Vitry (1291-1361), a musician from Vitry-en-Artois in France, called numbers of the form 2^m * 3^n "harmonic numbers". He asked if all powers of 2 and 3 differ by more than 1 except the pairs 1 and 2, 2 and 3, 3 and 4, 8 and 9 (which correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone). Levi Ben Gerson (1288-1344) answered yes by proving that 3^n +- 1 is not a power of 2 if n > 2; see A235365, A235366.

			8 + 1 = 2^3 + 1 = 3^2 = 9, so the pair 8 and 9 is in the sequence.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.

A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
H. Lenstra, Harmonic Numbers, MSRI, 1998.
J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
I. Peterson, Medieval Harmony, Math Trek, MAA, 2012.
Wikipedia, Gersonides
Wikipedia, Philippe de Vitry

Cf. A003586, A006899, A061987, A108906, A235365, A235366.

A002591 Largest prime factor of 3^(2n+1) - 1.

Original entry on oeis.org

2, 13, 11, 1093, 757, 3851, 797161, 4561, 34511, 363889, 368089, 1001523179, 391151, 8209, 20381027, 4404047, 2413941289, 2664097031, 17189128703, 797161, 86950696619, 380808546861411923, 927001, 96656723, 131713, 99810171997

Offset: 0

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 28.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 0..344
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Together with A274909 forms bisection of A074477.Cf. A057958, A059885, A085028, A133801, A235366.

Table[FactorInteger[3^(2n-1)-1][[-1,1]],{n,30}] (* Harvey P. Dale, Oct 19 2022 *)

a(n) = A074477(2n+1). - Max Alekseyev, May 22 2022

Corrected and extended by Jud McCranie, Jan 03 2001
Terms up to a(307) in b-file from Sean A. Irvine, Apr 20 2014
a(0) prepended and a(308)-a(344) added to b-file by Max Alekseyev, Apr 24 2019, Sep 10 2020, Aug 26 2021, May 22 2022

cp's OEIS Frontend

A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.

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

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A235365 Smallest odd prime factor of 3^n + 1, for n > 1.

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A108906 First differences of A006899.

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A218356 Minimal order of degree-n irreducible polynomials over GF(3).

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