A257997 -id:A257997 - cp's OEIS frontend

A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405

Offset: 1

Eric W. Weisstein

Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i*3^j*5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra.
Numbers k such that 8*k = EulerPhi(30*k). - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A165704: A165705(n) = A165704(a(n)). - Reinhard Zumkeller, Sep 26 2009
Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. - Hugo Pfoertner, Jul 16 2020
Also called ugly numbers, although it is not clear why. - Gus Wiseman, May 21 2021
Some woody bamboo species have extraordinarily long and stable flowering intervals that belong to this sequence. The model by Veller, Nowak & Davis justifies this observation from the evolutionary point of view. - Andrey Zabolotskiy, Jun 27 2021
Also those integers k for which, for every prime p > 5, p^(4*k) - 1 == 0 (mod 240*k). - Federico Provvedi, May 23 2022
As noted in the comments to A085152, Størmer's theorem implies that the only pairs of consecutive integers that appear as consecutive terms of this sequence are (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (15,16), (24,25), and (80,81). These all represent significant musical intervals. - Hal M. Switkay, Dec 05 2022

			From _Gus Wiseman_, May 21 2021: (Start)
The sequence of terms together with their prime indices begins:
      1: {}            25: {3,3}
      2: {1}           27: {2,2,2}
      3: {2}           30: {1,2,3}
      4: {1,1}         32: {1,1,1,1,1}
      5: {3}           36: {1,1,2,2}
      6: {1,2}         40: {1,1,1,3}
      8: {1,1,1}       45: {2,2,3}
      9: {2,2}         48: {1,1,1,1,2}
     10: {1,3}         50: {1,3,3}
     12: {1,1,2}       54: {1,2,2,2}
     15: {2,3}         60: {1,1,2,3}
     16: {1,1,1,1}     64: {1,1,1,1,1,1}
     18: {1,2,2}       72: {1,1,1,2,2}
     20: {1,1,3}       75: {2,3,3}
     24: {1,1,1,2}     80: {1,1,1,1,3}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Benoit Cloitre, Plot of abs(f(n)-s(n)) vs its mean values (blue) and vs loglog(n) (red).
M. J. Dominus, Infinite Lists in Perl.
Deborah Howard and Malcolm Longair, Harmonic Proportion and Palladio's "Quattro Libri", Journal of the Society of Architectural Historians (1982) 41 (2): 116-143.
Vaclav Kotesovec, Plot of a(n) / (exp((6*log(2)*log(3)*log(5)*n)^(1/3))/sqrt(30)) for n = 1..1200000
Rosetta Code, A collection of computer codes to compute 5-smooth numbers.
Raphael Schumacher, The Formula for the Distribution of the 3-Smooth Numbers, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv:1608.06928 [math.NT], 2016.
Sci.math, Ugly numbers.
Carl Veller, Martin A. Nowak and Charles C. Davis, Extended flowering intervals of bamboos evolved by discrete multiplication, Ecology Letters, 18 (2015), 653-659.
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Regular number.
Wikipedia, Talk:Regular number. Includes a discussion of the name.
Wikipedia, Størmer's theorem.

Subsequences: A003592, A003593, A051916 , A257997.
For p-smooth numbers with other values of p, see A003586, A002473, A051038, A080197, A080681, A080682, A080683.
Cf. A006530, A112757, A112758, A112759, A112763, A112764, A291719.
The partitions with these Heinz numbers are counted by A001399.
The conjugate opposite is A033942, counted by A004250.
The opposite is A059485, counted by A004250.
The non-3-smooth case is A080193, counted by A069905.
The conjugate is A037144, counted by A001399.
The complement is A279622, counted by A035300.
Requiring the sum of prime indices to be even gives A344297.
Cf. A000244, A002182, A002183, A035301, A056239, A112798, A261144, A344293.

import Data.Set (singleton, deleteFindMin, insert)
a051037 n = a051037_list !! (n-1)
a051037_list = f $ singleton 1 where
   f s = y : f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..500] | PrimeDivisors(n) subset [2,3,5]]; // Bruno Berselli, Sep 24 2012

A051037 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            numtheory[factorset](a) minus {2, 3,5 } ;
            if % = {} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A051037(n),n=1..100) ; # R. J. Mathar, Nov 05 2017

mx = 405; Sort@ Flatten@ Table[ 2^a*3^b*5^c, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}] (* Or *)
Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (* Robert G. Wilson v *)
With[{nn=10},Select[Union[Times@@@Flatten[Table[Tuples[{2,3,5},n],{n,0,nn}],1]],#<=2^nn&]] (* Harvey P. Dale, Feb 28 2022 *)

test(n)= {m=n; forprime(p=2,5, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,500,if(test(n),print1(n",")))

a(n)=local(m); if(n<1,0,n=a(n-1); until(if(m=n, forprime(p=2,5, while(m%p==0,m/=p)); m==1),n++); n)

list(lim)=my(v=List(),s,t); for(i=0,logint(lim\=1,5), t=5^i; for(j=0,logint(lim\t,3), s=t*3^j; while(s<=lim, listput(v,s); s<<=1))); Set(v) \\ Charles R Greathouse IV, Sep 21 2011; updated Sep 19 2016

smooth(P:vec,lim)={ my(v=List([1]),nxt=vector(#P,i,1),indx,t);
while(1, t=vecmin(vector(#P,i,v[nxt[i]]*P[i]),&indx);
if(t>lim,break); if(t>v[#v],listput(v,t)); nxt[indx]++);
Vec(v)
};
smooth([2,3,5], 1e4) \\ Charles R Greathouse IV, Dec 03 2013

is_A051037(n)=n<7||vecmax(factor(n,6)[, 1])<7 \\ M. F. Hasler, Jan 16 2015

def isok(n):
  while n & 1 == 0: n >>= 1
  while n % 3 == 0: n //= 3
  while n % 5 == 0: n //= 5
  return n == 1 #  Darío Clavijo, Dec 30 2022

from sympy import integer_log
def A051037(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,5)[0]+1):
            for j in range(integer_log(y:=x//5**i,3)[0]+1):
                c -= (y//3**j).bit_length()
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A051037gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5]
    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(A051037gen(), 65))) # Michael S. Branicky, Sep 17 2024

Let s(n) = Card(k | a(k)Benoit Cloitre, Dec 30 2001
The characteristic function of this sequence is given by:
  Sum_{n>=1} x^a(n) = Sum_{n>=1} -Möbius(30*n)*x^n/(1-x^n). - Paul D. Hanna, Sep 18 2011
a(n) = A143207(n) / 30. - Reinhard Zumkeller, Sep 13 2011
A204455(15*a(n)) = 15, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
A006530(a(n)) <= 5. - Reinhard Zumkeller, May 16 2015
Sum_{n>=1} 1/a(n) = Product_{primes p <= 5} p/(p-1) = (2*3*5)/(1*2*4) = 15/4. - Amiram Eldar, Sep 22 2020

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

A258023 Numbers of form (2^i)(3^j) or (3^i)(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 32, 36, 45, 48, 54, 64, 72, 75, 81, 96, 108, 125, 128, 135, 144, 162, 192, 216, 225, 243, 256, 288, 324, 375, 384, 405, 432, 486, 512, 576, 625, 648, 675, 729, 768, 864, 972, 1024, 1125, 1152, 1215, 1296

Offset: 1

Reinhard Zumkeller, May 16 2015

Union of A003586 and A003593;

A006530(a(n)) <= 5; A001221(a(n)) <= 2; a(n) mod 10 != 0.

			.   n |  a(n) |                 n |  a(n) |
. ----+-------+----------     ----+-------+------------
.   1 |    1  |  1             16 |   32  |  2^5
.   2 |    2  |  2             17 |   36  |  2^2 * 3^2
.   3 |    3  |  3             18 |   45  |  3^2 * 5
.   4 |    4  |  2^2           19 |   48  |  2^4 * 3
.   5 |    5  |  5             20 |   54  |  2 * 3^3
.   6 |    6  |  2 * 3         21 |   64  |  2^6
.   7 |    8  |  2^3           22 |   72  |  2^3 * 3^2
.   8 |    9  |  3^2           23 |   75  |  3 * 5^2
.   9 |   12  |  2^2 * 3       24 |   81  |  3^4
.  10 |   15  |  3 * 5         25 |   96  |  2^5 * 3
.  11 |   16  |  2^4           26 |  108  |  2^2 * 3^3
.  12 |   18  |  2 * 3^2       27 |  125  |  5^3
.  13 |   24  |  2^3 * 3       28 |  128  |  2^7
.  14 |   25  |  5^2           29 |  135  |  3^3 * 5
.  15 |   27  |  3^3           30 |  144  |  2^4 * 3^2

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (65000000 terms)

Cf. A003586, A003593, A051037, A006530, A001221, A010879, subsequence of: A051037, A257997, A337800, A337801.

import Data.List.Ordered (union)
a258023 n = a258023_list !! (n-1)
a258023_list = union a003586_list a003593_list

n = 10^4; Join[Table[2^i*3^j, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(10))) / sqrt(3). - Vaclav Kotesovec, Sep 22 2020

Sum_{n>=1} 1/a(n) = 27/8. - Amiram Eldar, Sep 23 2020

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A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.

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

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A258023 Numbers of form (2^i)*(3^j) or (3^i)*(5^j).

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A337800 Numbers of the form (2^i)*(3^j) or (2^i)*(5^j).

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A337801 Numbers of the form (2^i)*(5^j) or (3^i)*(5^j).

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A258023 Numbers of form (2^i)(3^j) or (3^i)(5^j).

A337800 Numbers of the form (2^i)(3^j) or (2^i)(5^j).

A337801 Numbers of the form (2^i)(5^j) or (3^i)(5^j).