A006899 -id:A006899 - cp's OEIS frontend

A108906 First differences of A006899.

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

A170803 Partial sums of A006899.

Original entry on oeis.org

1, 3, 6, 10, 18, 27, 43, 70, 102, 166, 247, 375, 618, 874, 1386, 2115, 3139, 5187, 7374, 11470, 18031, 26223, 42607, 62290, 95058, 154107, 219643, 350715, 527862, 790006, 1314294, 1845735, 2894311, 4488634, 6585786, 10780090, 15563059, 23951667, 38300574

Offset: 1

N. J. A. Sloane, Dec 12 2009

nonn

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

Cf. A006899, A170804, A170805.

a170803 n = a170803_list !! (n-1)
a170803_list = scanl1 (+) a006899_list -- Reinhard Zumkeller, Oct 09 2013

from sympy import integer_log
def A170803(n): return ((a:=1<>1) if 6**(k:=integer_log(m:=3**(n-1),6)[0])<<1>1)+(1<Chai Wah Wu, Feb 05 2025

A170804 Partial sums of (A006899, prefixed by a 1).

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 28, 44, 71, 103, 167, 248, 376, 619, 875, 1387, 2116, 3140, 5188, 7375, 11471, 18032, 26224, 42608, 62291, 95059, 154108, 219644, 350716, 527863, 790007, 1314295, 1845736, 2894312, 4488635, 6585787, 10780091, 15563060, 23951668, 38300575

Offset: 1

N. J. A. Sloane, Dec 12 2009

nonn

a(n) is the minimum number of three-in-a-rows passing through any cell in n-dimensional tic-tac-toe = the minimum value in the n-th row of A352419. - Ben Orlin, Mar 22 2022

Cf. A006899.
Equals A170803 + 1.
Minimum value in the n-th row of A352419.

from sympy import integer_log
def A170804(n): return 1 if n == 1 else ((a:=1<>1) if 6**(k:=integer_log(m:=3**(n-2),6)[0])<<1>1)+(1<Chai Wah Wu, Feb 05 2025

A085240 As exponents > 0 in powers of 2 or 3 (A006899) occur in A085238 exactly twice, a self-inverse permutation of natural numbers is induced.

Original entry on oeis.org

2, 1, 5, 7, 3, 10, 4, 12, 15, 6, 18, 8, 20, 23, 9, 25, 28, 11, 31, 13, 33, 36, 14, 38, 16, 41, 43, 17, 46, 49, 19, 51, 21, 54, 56, 22, 59, 24, 62, 64, 26, 67, 27, 69, 72, 29, 74, 77, 30, 80, 32, 82, 85, 34, 87, 35, 90, 93, 37, 95, 98, 39, 100, 40, 103, 105, 42, 108

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

Cf. A006899, A085238.

seq[lim_] := Module[{r2 = Range[0, Floor[Log2[lim]]], r3 = Range[0, Floor[Log[3, lim]]], s, m, v, ind, j}, s = Rest@Rest@ SortBy[Join[{#, 2^#} & /@ r2, {#, 3^#} & /@ r3], Last][[;; , 1]]; ind = Flatten /@ (Position[s, #] & /@ Range[Max[s]]); m = Max[Flatten[ind]]; v = Table[0, {m}]; Do[If[Length[i] == 2, v[[i[[1]]]] = i[[2]]; v[[i[[2]]]] = i[[1]]], {i, ind}]; TakeWhile[v, # > 0 &]]; seq[10^25] (* Amiram Eldar, Mar 25 2025 *)

A085238(n+1) = A085238(a(n)+1). [corrected by Amiram Eldar, Mar 25 2025]

a(a(n)) = n.

A374484 Index of A006899(n) in A003586.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 12, 13, 17, 19, 22, 27, 28, 34, 37, 41, 48, 49, 56, 62, 65, 74, 77, 84, 93, 95, 106, 111, 118, 130, 131, 143, 152, 157, 171, 175, 186, 199, 202, 218, 225, 235, 252, 253, 271, 281, 290, 309, 312, 329, 344, 350, 371, 378, 393, 413, 416, 439

Offset: 1

Chai Wah Wu, Sep 16 2024

Index of powers of 2 and 3 in 3-smooth numbers.

			A006899(10) = 64 which is the 17th term of A003586, therefore a(10) = 17.

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

Cf. A003586, A006899.

Disjoint union of A022330 and A022331.

seq[lim_] := Position[Times @@ IntegerExponent[#, {2, 3}] & /@ Sort[Flatten[ Table[2^i*3^j, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}] ]], 0] // Flatten; seq[10^11] (* Amiram Eldar, Sep 18 2024 *)

from sympy import integer_log
def A374484(n): return sum(((1<

A003586(a(n)) = A006899(n).

a(n) ~ c * n^2, where c = log(2)*log(3)/(2*(log(2) + log(3))^2) = 0.118598856384648... - Vaclav Kotesovec and Amiram Eldar, Sep 19 2024

A112989 Terms in A006899 which are closer to the next term than to the previous one.

Original entry on oeis.org

8, 27, 64, 243, 512, 2048, 6561, 16384, 59049, 131072, 524288, 1594323, 4194304, 14348907, 33554432, 129140163, 268435456, 1073741824, 3486784401, 8589934592, 31381059609, 68719476736, 274877906944, 847288609443, 2199023255552

Offset: 1

Zak Seidov, Jan 03 2006

nonn

Next terms are: 17592186044416, 68630377364883, 140737488355328, 562949953421312, 1853020188851841, 4503599627370496, 16677181699666569, 36028797018963968, 144115188075855872, 450283905890997363, 1152921504606846976. Is the sequence infinite? Cf. A006899 Numbers of the form 2^i or 3^j.

Cf. A006899.

un=Union[Flatten[Table[{2^n, 3^n}, {n, 60}]]];Do[If[un[[i]]-un[[i-1]]>un[[i+1]]-un[[i]], Print[un[[i]]]], {i, 2, Length[un]-1}]

A006899(n) - A006899(n-1) > A006899(n+1) - A006899(n)

A008472 Sum of the distinct primes dividing n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, 10, 2, 18, 16, 67, 19, 26, 14, 71, 5, 73

Offset: 1

Olivier Gérard

Sometimes called sopf(n).
Sum of primes dividing n (without repetition) (compare A001414).
Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7, ...]. - Gary W. Adamson, Feb 14 2008
Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008
a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009
a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009
a(A001043(n)) = A191583(n);
For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;
a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011
For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) + 2, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing mappings on a set with n elements.
a(n) + 3, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing partial mappings on a set with n elements.
(End)
The smallest m such that a(m) = n, or 0 if no such number m exists is A064502(n). The only integers that are not in the sequence are 1, 4 and 6. - Bernard Schott, Feb 07 2022

			a(18) = 5 because 18 = 2 * 3^2 and 2 + 3 = 5.
a(19) = 19 because 19 is prime.
a(20) = 7 because 20 = 2^2 * 5 and 2 + 5 = 7.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, On the asymptotic prime partitions of integers, arXiv:1609.06497 [math-ph], 2017.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

First difference of A024924.
Cf. A001414 (sopfr), A001222, A051731, A061397, A064502, A085020, A143535.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), this sequence (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

a008472 = sum . a027748_row  -- Reinhard Zumkeller, Mar 29 2012

[n eq 1 select 0 else &+[p[1]: p in Factorization(n)]: n in [1..100]]; // Vincenzo Librandi, Jun 24 2017

A008472 := n -> add(d, d = select(isprime, numtheory[divisors](n))):
seq(A008472(i), i = 1..40); # Peter Luschny, Jan 31 2012
A008472 := proc(n)
        add( d, d= numtheory[factorset](n)) ;
end proc: # R. J. Mathar, Jul 08 2012

Prepend[Array[Plus @@ First[Transpose[FactorInteger[#]]] &, 100, 2], 0]
Join[{0}, Rest[Total[Transpose[FactorInteger[#]][[1]]]&/@Range[100]]] (* Harvey P. Dale, Jun 18 2012 *)
(* Requires version 7.0+ *) Table[DivisorSum[n, # &, PrimeQ[#] &], {n, 75}] (* Alonso del Arte, Dec 13 2014 *)
Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

sopf(n) = local(fac=factor(n)); sum(i=1,matsize(fac)[1],fac[i,1])

vector(100,n,vecsum(factor(n)[,1]~)) \\ Derek Orr, May 13 2015

A008472(n)=vecsum(factor(n)[,1]) \\ M. F. Hasler, Jul 18 2015

from sympy import primefactors
def A008472(n): return sum(primefactors(n)) # Chai Wah Wu, Feb 03 2022

def A008472(n):
    return add(d for d in divisors(n) if is_prime(d))
print([A008472(i) for i in (1..40)]) # Peter Luschny, Jan 31 2012

[sum(prime_factors(n)) for n in range(1,74)] # Giuseppe Coppoletta, Jan 19 2015

Let n = Product_j prime(j)^k(j) where k(j) >= 1, then a(n) = Sum_j prime(j).
Additive with a(p^e) = p.
G.f.: Sum_{k >= 1} prime(k)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Dirichlet g.f.: primezeta(s-1)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p. - Wesley Ivan Hurt, Feb 04 2022
From Bernard Schott, Feb 07 2022: (Start)
For n > 0: a(A001020(n)) = 11, a(A001022(n)) = 13, a(A001026(n)) = 17, a(A001029(n)) = 19, a(A009967(n)) = 23, a(A009973(n)) = 29, a(A009975(n)) = 31, a(A009981(n)) = 37, a(A009985(n)) = 41, a(A009987(n)) = 43, a(A009991(n)) = 47.
For p odd prime, a(2*p) = p+2 <==> a(A100484(n)) = A052147(n) for n > 1. (End)
a(n) = Sum_{d|n} d * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

A033845 Numbers k of the form 2^i*3^j, where i and j >= 1.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664

Offset: 1

Jeff Burch

This sequence is easily confused with A003586, which gives numbers of the form 2^i*3^j with i, j >= 0, and is one-sixth of the present sequence. . 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
Solutions to phi(n)=n/3 [See J-M. de Koninck & A. Mercier, problème 733].
Numbers n such that Sum_{d prime divisor of n} 1/d = 5/6. - Benoit Cloitre, Apr 13 2002
Also n such that Sum_{d|n} mu(d)^2/d = 2. - Benoit Cloitre, Apr 15 2002
Complement of A006899 with respect to A003586. - Reinhard Zumkeller, Sep 25 2008
In the sieve of Eratosthenes, if one crosses numbers off multiple times, these numbers are crossed off twice, first for 2 and then for 3. - Alonso del Arte, Aug 22 2011
Subsequence of A051037. - Reinhard Zumkeller, Sep 13 2011
Numbers n such that Sum_{d|n} A008683(d)*A000041(d) = 7. - Carl Najafi, Oct 19 2011
Numbers n such that Sum_{d|n} A008683(d)*A000700(d) = 2. - Carl Najafi, Oct 20 2011
Solutions to the equation A001615(x) = 2x. - Enrique Pérez Herrero, Jan 02 2012
So these numbers are called Psi-perfect numbers [see J-M. de Koninck & A. Mercier, problème 654]. - Bernard Schott, Nov 20 2020

J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 733, page 94.
J-M. de Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, 2004, Problème 654, page 85.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A000423, A003586, A051037, A256617.
Cf. A001615, A006899, A086410, A086411, A008683, A143201.
Cf. A191475, A191476.

import Data.Set (singleton, deleteFindMin, insert)
a033845 n = a033845_list !! (n-1)
a033845_list = f (singleton (2*3)) where
   f s = m : f (insert (2*m) $ insert (3*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

mx = 12000; Sort@ Flatten@ Table[2^i*3^j, {i, Log[2, mx]}, {j, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(), N); for(n=0, log(lim\2)\log(3), N=6*3^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 02 2012

from sympy import integer_log
def A033845(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 6*bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A033845gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 6*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A033845gen(), 50))) # Michael S. Branicky, Sep 18 2024

Six times the 3-smooth numbers (A003586). - Ralf Stephan, Apr 16 2004
A086411(a(n)) - A086410(a(n)) = 1. - Reinhard Zumkeller, Sep 25 2008
A143201(a(n)) = 2. - Reinhard Zumkeller, Sep 13 2011
a(n) = 2^A191475(n) * 3^A191476(n). - Zak Seidov, Nov 01 2013
Sum_{n>=1} 1/a(n) = 1/2. - Amiram Eldar, Oct 13 2020

Edited by N. J. A. Sloane, Jan 31 2010 and May 26 2024.

A122437 Allowable values of the "dropping time" of the Collatz (3x+1) iteration.

Original entry on oeis.org

1, 3, 6, 8, 11, 13, 16, 19, 21, 24, 26, 29, 32, 34, 37, 39, 42, 44, 47, 50, 52, 55, 57, 60, 63, 65, 68, 70, 73, 75, 78, 81, 83, 86, 88, 91, 94, 96, 99, 101, 104, 106, 109, 112, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145, 148, 150, 153, 156, 158, 161

Offset: 1

T. D. Noe, Sep 06 2006

Only these numbers appear in A060445, which tabulates the "dropping time" of odd numbers. Note that all even numbers have a "dropping time" of 1.
a(n) is also the number of binary digits of 6^(n-1); for example, a(4)=8 since 6^(4-1)=216 in binary is 11011000, an 8-digit number. - Julio Cesar de la Yncera, Mar 28 2009
A positive integer (x) is an allowable value if and only if (x-1)/(1+log(2)/log(3)) - floor(x/(1+log(2)/log(3))) is not negative. - K. Spage, Oct 22 2009
Here the word "allowable" means that it is necessary for a sequence of iterates starting from odd value m to arrive at a value x = f^{floor(1+n+n*log(3)/log(2))}(m) < m, where n gives the number of odds in such a sequence including m, to have undergone precisely floor(1+n+n*log(3)/log(2)) iterations of f, where f(2*m)=m, f(2*m+1)=6*m+4. However, the formula for a(n+1) does not fully account for the order of odds and evens in such a sequence because it does not account for the effects of the "+1". Thus it is unknown whether it maximizes the value x for all values m. For example, fix m = 1 and the "+1" is enough to give the trivial cycle. So it is possible that for some m we have f^{floor(1+n+n*log(3)/log(2))}(m) >= m. - Jeffrey R. Goodwin, Aug 24 2011
The indices of the powers of 3 in A006899. - Ruud H.G. van Tol, Nov 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A022921 (number of 2^m between 3^n and 3^(n+1)), A122442 (least k having dropping time a(n)).

Cf. A006899.

Floor[1+Range[0,100]*(1+Log[2,3])] (* T. D. Noe, Sep 08 2006 *)
Map[Length[RealDigits[ #, 2][[1]]] &, Table[10^i, {i, 0, 50}]] (* Julio Cesar de la Yncera, Mar 28 2009 *)

a(n)=logint(3^(n-1),2)+n \\ Ruud H.G. van Tol, Nov 04 2022

a(1) = 1, a(n+1) = a(n) + A022921(n-1) + 1.
a(n+1) = floor(1 + n + n*log(3)/log(2)). - T. D. Noe, Sep 08 2006
a(n) = floor((1 + log(2)/log(3))*A020914(n-1)). - K. Spage, Oct 22 2009
a(n) = A020914(n-1) + n - 1. - K. Spage, Oct 23 2009 [corrected by Ruud H.G. van Tol, Nov 03 2022]
a(n) = a(n-1)+2 if 3^(n-1) < 2^(a(n-1)+2-(n-1)); a(n) = a(n-1)+3 otherwise. - V. Barbera, Aug 12 2025

Comment corrected and edited by Jon E. Schoenfield, Feb 27 2014

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).

cp's OEIS Frontend

A108906 First differences of A006899.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Extensions

A170803 Partial sums of A006899.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Python

A170804 Partial sums of (A006899, prefixed by a 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A085240 As exponents > 0 in powers of 2 or 3 (A006899) occur in A085238 exactly twice, a self-inverse permutation of natural numbers is induced.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A374484 Index of A006899(n) in A003586.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A112989 Terms in A006899 which are closer to the next term than to the previous one.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A008472 Sum of the distinct primes dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Sage

Sage