A108906 -id:A108906 - cp's OEIS frontend

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

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

A235366 Smallest odd prime factor of 3^n - 1.

Original entry on oeis.org

13, 5, 11, 7, 1093, 5, 13, 11, 23, 5, 797161, 547, 11, 5, 1871, 7, 1597, 5, 13, 23, 47, 5, 11, 398581, 13, 5, 59, 7, 683, 5, 13, 103, 11, 5, 13097927, 1597, 13, 5, 83, 7, 431, 5, 11, 47, 1223, 5, 491, 11, 13, 5, 107, 7, 11, 5, 13, 59, 14425532687, 5, 603901, 683, 13, 5, 11, 7, 221101, 5, 13, 11

Offset: 3

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 > 2, 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. - Sondow

One way to prove it is by the use of congruences. The powers of 3, modulo 80, are 3, 9, 27, 1, 3, 9, 27, 1, 3, 9, 27, 1, ... The powers of 2 are 2, 4, 8, 16, 32, 64, 48, 16, 32, 64, 48, 16, ... - Alonso del Arte, Jan 20 2014

			3^3 - 1 = 26 = 2 * 13, so a(3) = 13.
3^4 - 1 = 80 = 2^4 * 5, so a(4) = 5.
3^5 - 1 = 242 = 2 * 11^2, so a(5) = 11.

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

Max Alekseyev, Table of n, a(n) for n = 3..796 (terms to a(660) from Charles R Greathouse IV)
P. Franklin, Problem 2927, Amer. Math. Monthly, 30 (1923), p. 81.
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

See A235365 for 3^n + 1.

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

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

a(n)=factor(3^n>>valuation(3^n-1,2))[1,1] \\ Charles R Greathouse IV, Jan 20 2014

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

a(6+12n) = 7 as 3^(6+12n)-1 = (3^6)^(1+2n)-1 = 729^(1+2n)-1 = (728+1)^(1+2n)-1 == 1^(1+2n)-1 == 0 (mod 7), but 729^(1+2n)-1 = (730-1)^(1+2n)-1 == (-1)^(1+2n)-1 == -2 (mod 5).

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

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.

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

A364902 Let x, y be the greatest exponents of 2, 3 respectively such that 2^x, 3^y do not exceed n and let k_2, k_3 be n - 2^x, and n - 3^y respectively. Then for n such that k_2 = 0 or k_3 = 0, a(n) = n, else a(n) is the least novel number Min{pa(k_2), qa(k_3)}, where p, q are primes not equal to either 2 or 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 15, 8, 9, 7, 14, 20, 25, 35, 50, 16, 11, 22, 21, 28, 55, 70, 75, 40, 45, 49, 27, 13, 26, 33, 44, 32, 17, 34, 39, 52, 65, 98, 100, 56, 63, 77, 80, 121, 110, 105, 140, 112, 143, 154, 147, 196, 245, 135, 91, 130, 165, 220, 160, 85, 170, 195, 260, 64, 19, 38, 51, 68

Offset: 1

David James Sycamore and Michael De Vlieger Sep 21 2023

nonn

Motivated by the recursion D(2) known to reproduce A005940, this sequence uses a compound version based on a squarefree semiprime (6) rather than a prime, in which the terms are generated by a greedy algorithm related to the distances between n and the greatest powers of 2, and 3 not exceeding n. After a(9) = 9 each power of 2 or 3 is followed by the smallest prime not yet in the sequence. (e.g. 11 follows 16, 13 follows 27, etc).
There are no multiples of 6 in this sequence.
For k > 2, if a(i) = prime(k) = p and a(j) = p^2 then j-i is a term in A006899 (e.g. a(17) = 11, a(44) = 121 and 44 - 17 = 27 = 3^3).
Conjectures: (i). This is a permutation of A047253 with primes in order; (ii). All terms between consecutive prime terms, prime(k), prime(k+1) are prime(k)-smooth.

			a(n) = n for n <= 4 because all such n are powers of 2 or 3.
a(5) = least novel Min{a(1)*p,a(2)*q} = Min{p,2*q} for o,q prime != 2 or 3, so a(5) = 5.
17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
Data can be shown in tabular form in two distinct ways: First row starts with 1 and then rows start with a prime; alternatively each row starts with 2^i or 3^j:
 1;                       1;
 2;                       2;
 3,4;                     3;
 5,10,15,8,9;             4,5,10,15;
 7,14,20,25,35,50,16;     8;
 11,22,21,28,55...        9,7,14,20,25,35,50

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.

nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
Array[Set[{q[#1], p[#1],
      r[#1]}, {#1, #2,
        Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
       Prime[#2]}] & @@ {#, s[[#]]} &, w];
Do[If[n == 1,
   Set[{a[n], c[1]}, {1, True}],
   Array[Set[m[#], 1] &, w];
   Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
   Array[
    If[j[#] == 0,
      k[#] = n; flag = #,
      While[Set[k[#], Prime[m[#]] a[j[#]]];
       Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
   If[flag > 0,
    Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
    Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 24 2023 *)

For n > 6, a(A006899(n) + 1) = prime(n-2).

cp's OEIS Frontend

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

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A235366 Smallest odd prime factor of 3^n - 1.

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

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

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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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