A006899 -id:A006899 - cp's OEIS frontend

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

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

A085239 Sort the numbers 2^i and 3^j. Then a(n) is the base of the n-th term. Set a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

The density of 2's in this sequence is log(3)/log(6). The density of 3's in this sequence is log(2)/log(6). - Jennifer Buckley, Apr 24 2024

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

Cf. A000079, A000244.
Cf. A006899, A085238.
Cf. A152683, A152935.

a085239 1 = 1
a085239 n = a006899 n `mod` 2 + 2  -- Reinhard Zumkeller, Oct 09 2013

m = 40;
Join[{1}, If[Total[IntegerDigits[#, 2]] == 1, 2, 3]& /@ Union[3^Range[m], 2^Range[Length[IntegerDigits[3^m, 2]] - 1]]] (* Jean-François Alcover, Oct 07 2021 *)

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

from sympy import integer_log
def A085239(n): return 1 if n==1 else 2 if 6**integer_log(m:=3**(n-1),6)[0]<<1Chai Wah Wu, Feb 04 2025

A006899(n) = a(n)^A085238(n).

For n > 1: a(n) = 2 + A006899(n) mod 2. - Reinhard Zumkeller, Oct 09 2013

A086410 Smallest prime factor of 3-smooth numbers, with a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 18 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A006899, A020639, A033845, A086411.

Reap[For[n = 1, n <= 2*10^5, n++, If[EulerPhi[6*n] == 2*n, Sow[ FactorInteger[n][[1, 1]]]]]][[2, 1]] (* Jean-François Alcover, Sep 02 2016 *)

a(n) = A020639(A003586(n));
a(n) <= A086411(n) <= 3.
a(A033845(n)) = A086411(A033845(n))-1; a(A006899(n)) = A086411(A006899(n)). - Reinhard Zumkeller, Sep 25 2008

A086411 Greatest prime factor of 3-smooth numbers.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A006530, A006899, A033845, A086410.

Reap[Do[p = FactorInteger[n][[-1, 1]]; If[p < 5, Sow[p]], {n, 1, 2*10^5}] ][[2, 1]] (* Jean-François Alcover, Dec 17 2017 *)

a(n) = A006530(A003586(n)).
A086410(n) <= a(n) <= 3.
a(A033845(n)) = A086410(A033845(n))+1; a(A006899(n)) = A086410(A006899(n)). - Reinhard Zumkeller, Sep 25 2008
Conjecture: a(n) = A049237(n+1) for n>1. - R. J. Mathar, Jun 06 2024

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).

A306044 Powers of 2, 3 and 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 8388608

Offset: 1

Zak Seidov, Jun 18 2018

nonn

Union of A000079, A000244 and A000351.

Michel Marcus, Table of n, a(n) for n = 1..1370

Cf. A000079, A000244, A000351, A006899.

Cf. A226722, A226723, A226724.

N:= 10^7: # for terms <= N
sort(convert(`union`(seq({seq(b^i,i=0..ilog[b](N))},b=[2,3,5])),list)); # Robert Israel, Nov 18 2022

Union[2^Range[0, Log2[5^10]], 3^Range[Log[3, 5^10]], 5^Range[10]]

setunion(setunion(vector(logint(N=10^6,5)+1,k,5^(k-1)), vector(logint(N,3),k,3^k)), vector(logint(N,2),k,2^k)) \\ M. F. Hasler, Jun 24 2018

a(n)= my(f=[2,3,5],q=sum(k=1,#f,1/log(f[k]))); for(i=1,#f, my(p=logint(exp(n/q),f[i]),d=0,j=0,m=0); while(jRuud H.G. van Tol, Nov 16 2022 (with the help of the pari-users mailing list) Observation: with f=primes(P), d <= logint(P,2).

from sympy import integer_log
def A306044(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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-x.bit_length()-integer_log(x,3)[0]-integer_log(x,5)[0]
    return bisection(f,n,n) # Chai Wah Wu, Feb 05 2025

Sum_{n>=1} 1/a(n) = 11/4. - Amiram Eldar, Dec 10 2022

A098293 Powers of 2 alternating with powers of 3.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 8, 27, 16, 81, 32, 243, 64, 729, 128, 2187, 256, 6561, 512, 19683, 1024, 59049, 2048, 177147, 4096, 531441, 8192, 1594323, 16384, 4782969, 32768, 14348907, 65536, 43046721, 131072, 129140163, 262144, 387420489, 524288

Offset: 0

Wolfdieter Lang, Oct 18 2004

The finite sequence [1,2,3,4,9,8,27] is used in Timaios [35b] by Platon.

Luc Brisson, Le Même et l'Autre dans la Structure Ontologique du Timée de Platon, Klincksieck, Paris, 1974, p. 317.

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-6).

Except for initial 1, reordering of A006899.

&cat[ [2^n, 3^n]: n in [0..30]]; // Vincenzo Librandi, May 10 2015

seq(seq(k^n, k=2..3), n=0..19); # Zerinvary Lajos, Jun 29 2007

With[{nn=20},Riffle[2^Range[0,nn],3^Range[0,nn]]] (* Harvey P. Dale, Nov 28 2011 *)
Flatten[Table[{2^n, 3^n}, {n, 0, 20}]] (* Vincenzo Librandi, May 10 2015 *)

def A098293(n): return 3**(n>>1) if n&1 else 1<<(n>>1) # Chai Wah Wu, Sep 24 2024

a(2*k) = 2^k, a(2*k+1) = 3^k, k>=0.
G.f.: (1+x-3*x^2-2*x^3)/((1-2*x^2)*(1-3*x^2)).
a(n) = ((5-(-1)^n)/2)^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Dec 13 2014

A186927 Lesser of two consecutive 3-smooth numbers having no common divisors.

Original entry on oeis.org

1, 2, 3, 8, 27, 243, 2048, 524288, 129140163, 68630377364883, 36472996377170786403, 19342813113834066795298816, 706965049015104706497203195837614914543357369, 13703277223523221219433362313025801636536040755174924956117940937101787

Offset: 1

Charles R Greathouse IV and Reinhard Zumkeller, Mar 01 2011

nonn

a(n) = A003586(A186771(n)); A186928(n) = A003586(A186771(n) + 1).
Subsequence of A006899: all terms are either powers of 2 or of 3.
Najman improves an algorithm of Bauer & Bennett for computing the function that measures the minimal gap size f(k) in the sequence of integers at least one of whose prime factors exceeds k. This allows us to compute values of f(k) for larger k and obtain new values of f(k). - Jonathan Vos Post, Aug 18 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..21
M. Bauer and M. A. Bennett, Prime factors of consecutive integers, Mathematics of Computation 77 (2008), pp. 2455-2459.
Charles R Greathouse IV, Illustration of n, a(n) for n = 1..33
Filip Najman, Large strings of consecutive smooth integers, Aug 18, 2011

Cf. A186711.

smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; sn = smoothNumbers[3, 10^100]; Reap[For[i = 1, i <= Length[sn] - 1, i++, If[CoprimeQ[sn[[i]], sn[[i + 1]]], Sow[sn[[i]]]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2016 *)

A186928 Greater of two consecutive 3-smooth numbers having no common divisors.

Original entry on oeis.org

2, 3, 4, 9, 32, 256, 2187, 531441, 134217728, 70368744177664, 36893488147419103232, 19383245667680019896796723, 713623846352979940529142984724747568191373312, 13803492693581127574869511724554050904902217944340773110325048447598592

Offset: 1

Charles R Greathouse IV and Reinhard Zumkeller, Mar 01 2011

nonn

a(n) = A003586(A186771(n) + 1); A186927(n) = A003586(A186771(n));

also a subsequence of A006899: all terms are either powers of 2 or of 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..20

Subsequence of A006899.

Cf. A186711.

smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; sn = smoothNumbers[3, 10^100]; Reap[For[i = 1, i <= Length[sn] - 1, i++, If[CoprimeQ[sn[[i]], sn[[i + 1]]], Sow[sn[[i + 1]]]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2016 *)

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.

cp's OEIS Frontend

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

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A085239 Sort the numbers 2^i and 3^j. Then a(n) is the base of the n-th term. Set a(1)=1.

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A086410 Smallest prime factor of 3-smooth numbers, with a(1)=1.

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A086411 Greatest prime factor of 3-smooth numbers.

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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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A306044 Powers of 2, 3 and 5.

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A098293 Powers of 2 alternating with powers of 3.

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