A003591 -id:A003591 - cp's OEIS frontend

A165742 First differences of A003591.

1, 2, 3, 1, 6, 2, 12, 4, 17, 7, 8, 34, 14, 16, 68, 28, 32, 87, 49, 56, 64, 174, 98, 112, 128, 348, 196, 224, 256, 353, 343, 392, 448, 512, 706, 686, 784, 896, 1024, 1412, 1372, 1568, 1792, 2048, 423, 2401, 2744, 3136, 3584, 4096, 846, 4802, 5488, 6272, 7168, 8192, 1692, 9604, 10976, 12544, 14336, 2961, 13423, 3384, 19208

Offset: 1

Mick Purcell (mickpurcell(AT)gmail.com), Sep 25 2009

nonn

Conjecturally, the limit infimum is infinite. [Charles R Greathouse IV, Jun 28 2011]

			For n = 5, the 5th number of the form 2^i*7^j with i, j >= 0 is 8, and the 6th number of the form 2^i*7^j with i, j >= 0 is 14, so the difference is 14 - 8 = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

diff(v)=vector(#v-1,i,v[i+1]-v[i])
list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N<<=1));diff(vecsort(Vec(v))) \\ Charles R Greathouse IV, Jun 28 2011

A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192

Offset: 1

N. J. A. Sloane

Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.
Successive numbers k such that phi(210k) = 48k. - Artur Jasinski, Nov 05 2008
The divisors of 10! (A161466) are a finite subsequence. - Reinhard Zumkeller, Jun 10 2009
Numbers n such that A198487(n) > 0 and A107698(n) > 0. - Jaroslav Krizek, Nov 04 2011
A262401(a(n)) = a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers which are products of single-digit numbers. - N. J. A. Sloane, Jul 02 2017
Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - Richard Locke Peterson, May 09 2020
Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - Federico Provvedi, Jun 06 2022
The nonprimes with this property are all terms except for 2, 3, 5 and 7, i.e.: (1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, ...); the composite terms are all but the first one of this subsequence. ["Trivial" data provided mainly for search purpose.] - M. F. Hasler, Jun 06 2023

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5841 terms from N. J. A. Sloane)
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.
University of Ulm, The first 5842 terms.
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Smooth number

Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.
Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.
Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.
Cf. A262401.

import Data.Set (singleton, deleteFindMin, fromList, union)
a002473 n = a002473_list !! (n-1)
a002473_list = f $ singleton 1 where
   f s = x : f (s' `union` fromList (map (* x) [2,3,5,7]))
         where (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Mar 08 2014, Apr 02 2012, Apr 01 2012

[n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // Bruno Berselli, Sep 24 2012

Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&]
aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* Artur Jasinski, Nov 05 2008 *)
mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* Harvey P. Dale, Aug 13 2012 *)
mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A002473(n)=n<11||vecmax(factor(n,8)[,1])<8 \\ M. F. Hasler, Jan 16 2015

list(lim)=my(v=List(),t); for(a=0,logint(lim\1,7), for(b=0,logint(lim\7^a,5), for(c=0,logint(lim\7^a\5^b,3), t=3^c*5^b*7^a; while(t<=lim, listput(v,t); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

import heapq
from itertools import islice
from sympy import primerange
def A002473gen(p=7): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    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(A002473gen(), 65))) # Michael S. Branicky, Nov 19 2022

from sympy import integer_log
def A002473(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):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            i7 = 7**i
            m = x//i7
            for j in range(integer_log(m,5)[0]+1):
                j5 = 5**j
                r = m//j5
                for k in range(integer_log(r,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

A006530(a(n)) <= 7. - Reinhard Zumkeller, Apr 01 2012

Sum_{n>=1} 1/a(n) = Product_{primes p <= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - Amiram Eldar, Sep 22 2020

More terms from James Sellers, Dec 23 1999
Additional comments from Michel Lecomte, Jun 09 2007
Edited by M. F. Hasler, Jan 16 2015

A003592 Numbers of the form 2^i*5^j with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000, 2048, 2500, 2560, 3125, 3200, 4000, 4096, 5000, 5120, 6250, 6400, 8000, 8192, 10000, 10240, 12500, 12800

Offset: 1

N. J. A. Sloane

These are the natural numbers whose reciprocals are terminating decimals. - David Wasserman, Feb 26 2002
A132726(a(n), k) = 0 for k <= a(n); A051626(a(n)) = 0; A132740(a(n)) = 1; A132741(a(n)) = a(n). - Reinhard Zumkeller, Aug 27 2007
Where record values greater than 1 occur in A165706: A165707(n) = A165706(a(n)). - Reinhard Zumkeller, Sep 26 2009
Also numbers that are divisible by neither 10k - 7, 10k - 3, 10k - 1 nor 10k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010
A204455(5*a(n)) = 5, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
Since p = 2 and q = 5 are coprime, sum_{n >= 1} 1/a(n) = sum_{i >= 0} sum_{j >= 0} 1/p^i * 1/q^j = sum_{i >= 0} 1/p^i q/(q - 1) = p*q/((p-1)*(q-1)) = 2*5/(1*4) = 2.5. - Franklin T. Adams-Watters, Jul 07 2014
Conjecture: Each positive integer n not among 1, 4 and 12 can be written as a sum of finitely many numbers of the form 2^a*5^b + 1 (a,b >= 0) with no one dividing another. This has been verified for n <= 3700. - Zhi-Wei Sun, Apr 18 2023
1,2 and 4,5 are the only consecutive terms in the sequence. - Robin Jones, May 03 2025

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 73.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (200000 terms)
Math Stack Exchange, Are (1,2) and (4,5) the only two consecutive pairs in A003592, the integers of the form 2^i5^j?
Eric Weisstein's World of Mathematics, Regular Number
Eric Weisstein's World of Mathematics, Decimal Expansion

Complement of A085837. Cf. A094958, A022333 (list of j), A022332 (list of i).
Cf. A003586, A003591, A003593, A003594, A003595, A257997.
Cf. A164768 (difference between consecutive terms)

Filtered([1..10000],n->PowerMod(10,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003592 n = a003592_list !! (n-1)
a003592_list = f $ singleton 1 where
   f s = y : f (insert (2 * y) $ insert (5 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..10000] | PrimeDivisors(n) subset [2,5]]; // Bruno Berselli, Sep 24 2012

isA003592 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {2,5} = {} );
    end if;
end proc:
A003592 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA003592(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 16 2012

twoFiveableQ[n_] := PowerMod[10, n, n] == 0; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Jan 12 2012 *)
twoFiveableQ[n_] := Union[ MemberQ[{1, 3, 7, 9}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 10]] == {False}; twoFiveableQ[1] = True; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Oct 26 2010 *)
maxExpo = 14; Sort@ Flatten@ Table[2^i * 5^j, {i, 0, maxExpo}, {j, 0, Log[5, 2^(maxExpo - i)]}] (* Or *)
Union@ Flatten@ NestList[{2#, 4#, 5#} &, 1, 7] (* Robert G. Wilson v, Apr 16 2011 *)

list(lim)=my(v=List(),N);for(n=0,log(lim+.5)\log(5),N=5^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

# A003592.py
from heapq import heappush, heappop
def A003592():
    pq = [1]
    seen = set(pq)
    while True:
        value = heappop(pq)
        yield value
        seen.remove(value)
        for x in 2*value, 5*value:
            if x not in seen:
                heappush(pq, x)
                seen.add(x)
sequence = A003592()
A003592_list = [next(sequence) for _ in range(100)]

from sympy import integer_log
def A003592(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-sum((x//5**i).bit_length() for i in range(integer_log(x,5)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

def isA003592(n) :
    return not any(d != 2 and d != 5 for d in prime_divisors(n))
@CachedFunction
def A003592(n) :
    if n == 1 : return 1
    k = A003592(n-1) + 1
    while not isA003592(k) : k += 1
    return k
[A003592(n) for n in (1..48)]  # Peter Luschny, Jul 20 2012

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(10*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
a(n) ~ exp(sqrt(2*log(2)*log(5)*n)) / sqrt(10). - Vaclav Kotesovec, Sep 22 2020
a(n) = 2^A022332(n) * 5^A022333(n). - R. J. Mathar, Jul 06 2025

Incomplete Python program removed by David Radcliffe, Jun 27 2016

A003594 Numbers of the form 3^i*7^j with i, j >= 0.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243, 343, 441, 567, 729, 1029, 1323, 1701, 2187, 2401, 3087, 3969, 5103, 6561, 7203, 9261, 11907, 15309, 16807, 19683, 21609, 27783, 35721, 45927, 50421, 59049, 64827, 83349, 107163, 117649

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
Vaclav Kotesovec, Graph - the asymptotic ratio (600000 terms).

Cf. A003586, A003591, A003592, A003593, A003595.

Cf. A025642, A025665.

Filtered([1..120000],n->PowerMod(21,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003594 n = a003594_list !! (n-1)
a003594_list = f $ singleton 1 where
   f s = y : f (insert (3 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..120000] | PrimeDivisors(n) subset [3,7]]; // Bruno Berselli, Sep 24 2012

f[upto_]:=Sort[Select[Flatten[3^First[#] 7^Last[#] & /@ Tuples[{Range[0, Floor[Log[3, upto]]], Range[0, Floor[Log[7, upto]]]}]], # <= upto &]]; f[120000]  (* Harvey P. Dale, Mar 04 2011 *)
fQ[n_] := PowerMod[21, n, n] == 0; Select[Range[120000], fQ] (* Bruno Berselli, Sep 24 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003594(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(integer_log(x//7**i,3)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(21*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
Sum_{n>=1} 1/a(n) = (3*7)/((3-1)*(7-1)) = 7/4. - Amiram Eldar, Sep 22 2020
a(n) ~ exp(sqrt(2*log(3)*log(7)*n)) / sqrt(21). - Vaclav Kotesovec, Sep 22 2020
a(n) = 3^A025642(n) * 7^A025665(n). - R. J. Mathar, Jul 06 2025

A108090 Numbers of the form (11^i)*(13^j).

Original entry on oeis.org

1, 11, 13, 121, 143, 169, 1331, 1573, 1859, 2197, 14641, 17303, 20449, 24167, 28561, 161051, 190333, 224939, 265837, 314171, 371293, 1771561, 2093663, 2474329, 2924207, 3455881, 4084223, 4826809, 19487171, 23030293, 27217619

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.

import Data.Set (singleton, deleteFindMin, insert)
a108090 n = a108090_list !! (n-1)
a108090_list = f $ singleton (1,0,0) where
   f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // Vincenzo Librandi, Jun 27 2016

mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),t); for(j=0,logint(lim\=1,13), t=13^j; while(t<=lim, listput(v,t); t*=11)); Set(v) \\ Charles R Greathouse IV, Aug 29 2016

from sympy import integer_log
def A108090(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-sum(integer_log(x//13**i,11)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Sum_{n>=1} 1/a(n) = (11*13)/((11-1)*(13-1)) = 143/120. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - Vaclav Kotesovec, Sep 23 2020

A109720 Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1

Offset: 0

Bruce Corrigan (scentman(AT)myfamily.com), Aug 09 2005

This sequence also represents n^12 mod 7; n^18 mod 7; (exponents are = 0 mod 6).
Characteristic sequence for numbers n>=1 to be relatively prime to 7. - Wolfdieter Lang, Oct 29 2008
a(n+4), n>=0, (periodic 1,1,1,0,1,1,1) is also the characteristic sequence for mod m reduced positive odd numbers (i.e., gcd(2*n+1,m)=1, n>=0) for each modulus m from 7*A003591 = [7,14,28,49,56,98,112,196,...]. [Wolfdieter Lang, Feb 04 2012]

Index entries for characteristic functions - _Reinhard Zumkeller_, Nov 30 2009
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1)

Cf. A010876 = n mod 7; A053879 = n^2 mod 7; A070472 = m^3 mod 7; A070512 = n^4 mod 7; A070593 = n^5 mod 7.

Cf. A168185, A145568, A168184, A168182, A168181, A097325, A011558, A166486, A011655, A000035.

PadRight[{},120,{0,1,1,1,1,1,1}] (* Harvey P. Dale, Jul 09 2018 *)

a(n)=n^6%7 \\ Charles R Greathouse IV, Sep 24 2015

[power_mod(n,6,7)for n in range(0, 105)] # Zerinvary Lajos, Nov 06 2009

a(n) = 0 if n=0 mod 7; a(n)= 1 else.
G.f. = (x+x^2+x^3+x^4+x^5+x^6)/(1-x^7)= -x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( (x-1)*(1+x+x^2+x^3+x^4+x^5+x^6) ).
a(n)=1-A082784(n); a(A047304(n))=1; a(A008589(n))=0; A033439(n) = SUM(a(k)*(n-k): 0<=k<=n). - Reinhard Zumkeller, Nov 30 2009
Multiplicative with a(p) = (if p=7 then 0 else 1), p prime. - Reinhard Zumkeller, Nov 30 2009
Dirichlet g.f. (1-7^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

A107326 Numbers of the form (2^i)*(13^j).

Original entry on oeis.org

1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 128, 169, 208, 256, 338, 416, 512, 676, 832, 1024, 1352, 1664, 2048, 2197, 2704, 3328, 4096, 4394, 5408, 6656, 8192, 8788, 10816, 13312, 16384, 17576, 21632, 26624, 28561, 32768, 35152, 43264, 53248, 57122

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 21 2005

A204455(13*a(n)) = 13, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Vincenzo Librandi)

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599.

fQ[n_] := PowerMod[26,n,n]==0; Select[Range[60000],fQ] (* Vincenzo Librandi, Feb 04 2012 *)
mx = 60000; Sort@ Flatten@ Table[2^i*13^j, {i, 0, Log[2, mx]}, {j, 0, Log[13, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(),N); for(n=0,log(lim)\log(13),N=13^n; while(N<=lim,listput(v,N);N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Sum_{n>=1} 1/a(n) = (2*13)/((2-1)*(13-1)) = 13/6. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(2)*log(13)*n)) / sqrt(26). - Vaclav Kotesovec, Sep 23 2020

A107364 Numbers of the form (3^i)*(13^j).

Original entry on oeis.org

1, 3, 9, 13, 27, 39, 81, 117, 169, 243, 351, 507, 729, 1053, 1521, 2187, 2197, 3159, 4563, 6561, 6591, 9477, 13689, 19683, 19773, 28431, 28561, 41067, 59049, 59319, 85293, 85683, 123201, 177147, 177957, 255879, 257049, 369603, 371293, 531441

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 23 2005

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326.

[n: n in [1..10^7] | PrimeDivisors(n) subset [3, 13]]; // Vincenzo Librandi, Jun 27 2016

mx = 540000; Sort@ Flatten@ Table[3^i*13^j, {i, 0, Log[3, mx]}, {j, 0, Log[13, mx/3^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
fQ[n_]:=PowerMod[39, n, n] == 0; Select[Range[2 10^7], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Sum_{n>=1} 1/a(n) = (3*13)/((3-1)*(13-1)) = 13/8. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(3)*log(13)*n)) / sqrt(39). - Vaclav Kotesovec, Sep 23 2020

A036566 Numbers of form 7^i*8^j with i, j >= 0, sorted.

Original entry on oeis.org

1, 7, 8, 49, 56, 64, 343, 392, 448, 512, 2401, 2744, 3136, 3584, 4096, 16807, 19208, 21952, 25088, 28672, 32768, 117649, 134456, 153664, 175616, 200704, 229376, 262144, 823543, 941192, 1075648, 1229312, 1404928, 1605632, 1835008, 2097152

Offset: 1

David W. Wilson

Could be rearranged as a triangle of numbers in which i-th row is {7^(i-j)*8^j, 0<=j<=i}; i >= 0. (This would produce a different sequence, of course).

The sum of the reciprocals of the terms of this sequence is equal to 4/3. Brief proof: as gcd(7, 8) = 1, 1 + 1/7 + 1/8 + 1/49 + 1/56 + 1/64 + 1/343 + ... = (Sum_{k>=0} 1/7^k) * (Sum_{m>=0} 1/8^m) = (1/(1-1/7)) * (1/(1-1/8)) = (7/(7-1)) * (8/(8-1)) = 4/3. - Bernard Schott, Oct 24 2019

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Subsequence of A003591.

N:= 10^7: # for all terms <= N
sort([seq(seq(7^i*8^j,j=0..floor(log[8](N/7^i))),i=0..floor(log[7](N)))]); # Robert Israel, Oct 24 2019

n = 10^6; Flatten[Table[7^i*8^j, {i, 0, Log[7, n]}, {j, 0, Log[8, n/7^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

a(n) ~ exp(sqrt(2*log(7)*log(8)*n)) / sqrt(56). - Vaclav Kotesovec, Sep 25 2020

a(n) = 7^A025669(n) * 8^A025675(n). - R. J. Mathar, Jul 06 2025

A107466 Numbers of the form (5^i)*(13^j).

Original entry on oeis.org

1, 5, 13, 25, 65, 125, 169, 325, 625, 845, 1625, 2197, 3125, 4225, 8125, 10985, 15625, 21125, 28561, 40625, 54925, 78125, 105625, 142805, 203125, 274625, 371293, 390625, 528125, 714025, 1015625, 1373125, 1856465, 1953125, 2640625

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), May 27 2005

nonn

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

Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364.

mx = 2700000; Sort@ Flatten@ Table[5^i*13^j, {i, 0, Log[5, mx]}, {j, 0, Log[13, mx/5^i]}] (* Robert G. Wilson v, Aug 17 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(13),N=13^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A107466(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-sum(integer_log(x//13**i,5)[0]+1 for i in range(integer_log(x,13)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Sum_{n>=1} 1/a(n) = (5*13)/((5-1)*(13-1)) = 65/48. - Amiram Eldar, Sep 23 2020

a(n) ~ exp(sqrt(2*log(5)*log(13)*n)) / sqrt(65). - Vaclav Kotesovec, Sep 23 2020

cp's OEIS Frontend

A165742 First differences of A003591.

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

PARI

Python

Python

Formula

Extensions

A003592 Numbers of the form 2^i*5^j with i, j >= 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Python

Python

Sage

Formula

Extensions

A003594 Numbers of the form 3^i*7^j with i, j >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Haskell

Magma

Mathematica

PARI

Python

Formula

A108090 Numbers of the form (11^i)*(13^j).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Formula

A109720 Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.

Views

Author

Keywords