A051037 -id:A051037 - cp's OEIS frontend

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 117, 120

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v*13^w with r, s, t, u, v, w >= 0.

			33 = 3*11 and 39 = 3*13 are terms but 34 = 2*17 is not.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A080196. For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080681, A080682, A080683.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(13)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n, {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)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,13, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A080197(n,p=13)=n<=p||vecmax(factor(n,p+1)[,1])<=p \\ M. F. Hasler, Jan 16 2015

list(lim,p=13)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=13): # 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(agen(), 69))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080197(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 g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,13)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 13} p/(p-1) = (2*3*5*7*11*13)/(1*2*4*6*10*12) = 1001/192. - Amiram Eldar, Sep 22 2020

A266755 Expansion of 1/((1-x^2)(1-x^3)(1-x^4)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120, 114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169

Offset: 0

N. J. A. Sloane, Jan 10 2016

This is the same as A005044 but without the three leading zeros. There are so many situations where one wants this sequence rather than A005044 that it seems appropriate for it to have its own entry.
But see A005044 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_3.
The Molien series for the finite Coxeter group of type D_k (k >= 3) has g.f. = 1/Product_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
Also, Molien series for invariants of finite Coxeter group A_3. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k.
a(n) is the number of partitions of n into parts 2, 3, and 4. - Joerg Arndt, Apr 16 2017
From Gus Wiseman, May 23 2021: (Start)
Also the number of integer partitions of n into at most n/2 parts, none greater than 3. The case of any maximum is A110618. The case of any length is A001399. The Heinz numbers of these partitions are given by A344293.
For example, the a(2) = 1 through a(13) = 5 partitions are:
    2  3  22  32  33   322  332   333   3322   3332   3333    33322
          31      222  331  2222  3222  3331   32222  33222   33331
                  321       3221  3321  22222  33221  33321   322222
                            3311        32221  33311  222222  332221
                                        33211         322221  333211
                                                      332211
                                                      333111
(End)

			G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + ... - _Michael Somos_, Jan 29 2022

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.
Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.
A variant of A005044.
Cf. A001400 (partial sums).
Cf. A308065.
Number of partitions of n whose Heinz number is in A344293.
A001399 counts partitions with all parts <= 3, ranked by A051037.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A110618 counts partitions of n into at most n/2 parts, ranked by A344291.
Cf. A000041, A008642, A279622, A325691, A344294, A344297.

I:=[1,0,1,1,2,1,3,2,4]; [n le 9 select I[n] else Self(n-2)+ Self(n-3)+Self(n-4)-Self(n-5)-Self(n-6)-Self(n-7)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Jan 11 2016

CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^4)), {x, 0, 100}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1}, {1,0,1,1,2,1,3,2,4}, 100] (* Vincenzo Librandi, Jan 11 2016 *)
Table[Length[Select[IntegerPartitions[n],Length[#]<=n/2&&Max@@#<=3&]],{n,0,30}] (* Gus Wiseman, May 23 2021 *)
a[ n_] := Round[(n + 3*(2 - Mod[n,2]))^2/48]; (* Michael Somos, Jan 29 2022 *)

Vec(1/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jan 11 2016

{a(n) = round((n + 3*(2-n%2))^2/48)}; /* Michael Somos, Jan 29 2022 */

(1/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. - Vincenzo Librandi, Jan 11 2016
a(n) = a(-9-n) for all n in Z. a(n) = a(n+3) for all n in 2Z. - Michael Somos, Jan 29 2022
E.g.f.: exp(-x)*(81 - 18*x + exp(2*x)*(107 + 60*x + 6*x^2) + 64*exp(x/2)*cos(sqrt(3)*x/2) + 36*exp(x)*(cos(x) - sin(x)))/288. - Stefano Spezia, Mar 05 2023
For n >= 3, if n is even, a(n) = a(n-3) + floor(n/4) + 1, otherwise a(n) = a(n-3). - Robert FERREOL, Feb 05 2024
a(n) = floor((n^2+9*n+(3*n+9)*(-1)^n+39)/48). - Hoang Xuan Thanh, Jun 03 2025

A159991 Powers of 60: a(n) = 60^n.

Original entry on oeis.org

1, 60, 3600, 216000, 12960000, 777600000, 46656000000, 2799360000000, 167961600000000, 10077696000000000, 604661760000000000, 36279705600000000000, 2176782336000000000000, 130606940160000000000000, 7836416409600000000000000, 470184984576000000000000000

Offset: 0

Reinhard Zumkeller, May 01 2009

			G.f. = 1 + 60*x + 3600*x^2 + 216000*x^3 + 12960000*x^4 + 77600000*x^5 + ... - _Michael Somos_, Jan 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Wikipedia, Sexagesimal
Index entries for linear recurrences with constant coefficients, signature (60).

Subsequence of A051037.

Cf. A159990, A159993, A159995, A088157, A091649, A125628, A091720, A091721, A091722, A060707, A070197.

List([0..20],n->60^n); # Muniru A Asiru, Nov 21 2018

[60^n: n in [0..20]]; // Vincenzo Librandi, May 02 2011

[60^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

60^Range[0,15] (* Harvey P. Dale, Jun 02 2011 *)

A159991(n):=60^n$
makelist(A159991(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=60^n \\ Charles R Greathouse IV, May 02 2011

a(n)=60^n \\ Charles R Greathouse IV, Jun 19 2015

powers(60,8) \\ Charles R Greathouse IV, Jun 19 2015

for n in range(0,20): print(60**n, end=', ') # Stefano Spezia, Nov 21 2018

a(n) = A000400(n)*A011557(n) = A000351(n)*A001021(n) = A000302(n)*A001024(n) = A000244(n)*A009964(n). (Corrected by Robert B Fowler, Jan 25 2023)
From Muniru A Asiru, Nov 21 2018: (Start)
a(n) = 60^n.
a(n) = 60*a(n-1) for n > 0, a(0) = 1.
G.f.: 1/(1-60*x).
E.g.f: exp(60*x). (End)
a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 01 2019

A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits, leading zeros omitted.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 0

Henry Bottomley, Jun 06 2000

From Wolfdieter Lang, Jan 16 2018: (Start)
The symbols used for 0..9 in this base 60 notation are 00, 01, ..., 09, but leading zeros are omitted.
For the Sumerian-Babylonian sexagesimal-decimal number system which uses two positions for each base-60 position filled with only one-digit numbers alternating between ranges of 0 to 9 and 0 to 5 see the link below.
(End)
For n < 1440, US and NATO military time designation of n minutes since midnight. - J. Lowell, Dec 29 2020

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.
Georges Ifrah, Histoire Universelle des Chiffres, Paris, 1981.
Georges Ifrah, From one to zero, A universal history of numbers, Viking Penguin Inc., 1985.
Georges Ifrah, Universalgeschichte der Zahlen, Campus Verlag, Frankfurt, New York, 2. Auflage, 1987, pp. 210-221.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Wolfdieter Lang, Sumerian-Babylonian sexagesimal-decimal number system.

Cf. A049872, A131650,
Note also that A250073 = a(A000079(n)), A250089 = a(A051037(n)), A254334 = a(A000244(n)), A254335 = a(A000351(n)), A254336 = a(A011557(n)).
See also A281863 (value of the 0,1,2,...,n-th digit of a(n), counted from the right), A282622 (length of a(n), #digits, for n >= 1).

Array[FromDigits@ Apply[Join, PadLeft[#, 2] & /@ IntegerDigits@ IntegerDigits[#, 60]] &, 71, 0] (* Michael De Vlieger, Jan 11 2018 *)

A055643(n)=fromdigits(digits(n,60),100) \\ M. F. Hasler, Jan 09 2018

def a(n): return n if n < 60 else 100*a(n//60) + n%60
print([a(n) for n in range(71)]) # Michael S. Branicky, Oct 22 2022

a(60*n+r) = 100*a(n) + r, 0 <= r <= 59. - Jianing Song, Oct 22 2022

a(69) and a(70) from WG Zeist, Sep 08 2012

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

A003596 Numbers of the form 2^i * 11^j.

Original entry on oeis.org

1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 121, 128, 176, 242, 256, 352, 484, 512, 704, 968, 1024, 1331, 1408, 1936, 2048, 2662, 2816, 3872, 4096, 5324, 5632, 7744, 8192, 10648, 11264, 14641, 15488, 16384, 21296, 22528, 29282, 30976, 32768

Offset: 1

N. J. A. Sloane

A204455(11*a(n)) = 11, and only for these numbers. - Wolfdieter Lang, Feb 04 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 100 terms from Vincenzo Librandi)

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

Filtered([1..33000],n->PowerMod(22,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003596 n = a003596_list !! (n-1)
a003596_list = f $ singleton (1,0,0) where
   f s = y : f (insert (2 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..2*10^5] | PrimeDivisors(n) subset [2, 11]]; // Vincenzo Librandi, Jun 27 2016

fQ[n_] := PowerMod[22,n,n]==0; Select[Range[40000], fQ] (* Vincenzo Librandi, Feb 04 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

from sympy import integer_log
def A003596(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//11**i).bit_length() for i in range(integer_log(x,11)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(22*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) = (2*11)/((2-1)*(11-1)) = 11/5. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(2)*log(11)*n)) / sqrt(22). - Vaclav Kotesovec, Sep 23 2020

A003598 Numbers of the form 5^i * 11^j.

Original entry on oeis.org

1, 5, 11, 25, 55, 121, 125, 275, 605, 625, 1331, 1375, 3025, 3125, 6655, 6875, 14641, 15125, 15625, 33275, 34375, 73205, 75625, 78125, 161051, 166375, 171875, 366025, 378125, 390625, 805255, 831875, 859375, 1771561, 1830125, 1890625

Offset: 1

N. J. A. Sloane

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

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A108698, A003599, A107788, A108687, A108779, A108090.

Filtered([1..2*10^6],n->PowerMod(55,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003598 n = a003598_list !! (n-1)
a003598_list = f $ singleton (1,0,0) where
   f s = y : f (insert (5 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..2*10^6] | PrimeDivisors(n) subset [5, 11]]; // Vincenzo Librandi, Jun 27 2016

Take[Union[(5^#[[1]] 11^#[[2]])&/@Tuples[Range[0,20],{2}]],50]  (* Harvey P. Dale, Dec 26 2010 *)
fQ[n_]:=PowerMod[55, n, n] == 0; Select[Range[2*10^6], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N*=5));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

[n for n in (1..2*10^6) if n%55 in {0, 1, 5, 11, 15, 20, 25, 45} and all(x in {5,11} for x in prime_factors(n))] # F. Chapoton, Mar 16 2020

An asymptotic formula for a(n) is roughly 1/sqrt(55)*exp(sqrt(2*log(5)*log(11)*n)). - Benoit Cloitre, Mar 08 2002
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(55*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) = (5*11)/((5-1)*(11-1)) = 11/8. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(5)*log(11)*n)) / sqrt(55). - Vaclav Kotesovec, Sep 23 2020

A003597 Numbers of the form 3^i*11^j.

Original entry on oeis.org

1, 3, 9, 11, 27, 33, 81, 99, 121, 243, 297, 363, 729, 891, 1089, 1331, 2187, 2673, 3267, 3993, 6561, 8019, 9801, 11979, 14641, 19683, 24057, 29403, 35937, 43923, 59049, 72171, 88209, 107811, 131769, 161051, 177147, 216513, 264627, 323433

Offset: 1

N. J. A. Sloane

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

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

Filtered([1..324000],n->PowerMod(33,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003597 n = a003597_list !! (n-1)
a003597_list = f $ singleton (1,0,0) where
   f s = y : f (insert (3 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..4*10^5] | PrimeDivisors(n) subset [3, 11]]; // Vincenzo Librandi, Jun 27 2016

fQ[n_]:=PowerMod[33, n, n] == 0; Select[Range[4*10^5], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(33*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*11)/((3-1)*(11-1)) = 33/20. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(3)*log(11)*n)) / sqrt(33). - Vaclav Kotesovec, Sep 23 2020

A003599 Numbers of the form 7^i*11^j.

Original entry on oeis.org

1, 7, 11, 49, 77, 121, 343, 539, 847, 1331, 2401, 3773, 5929, 9317, 14641, 16807, 26411, 41503, 65219, 102487, 117649, 161051, 184877, 290521, 456533, 717409, 823543, 1127357, 1294139, 1771561, 2033647, 3195731, 5021863, 5764801

Offset: 1

N. J. A. Sloane

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

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A107788, A108687, A108779, A108090.

import Data.Set (singleton, deleteFindMin, insert)
a003599 n = a003599_list !! (n-1)
a003599_list = f $ singleton (1,0,0) where
   f s = y : f (insert (7 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

[n: n in [1..6*10^6] | PrimeDivisors(n) subset [7, 11]]; // Vincenzo Librandi, Jun 27 2016

Take[Union[7^#[[1]] 11^#[[2]]&/@Tuples[Range[0,9],2]],40] (* Harvey P. Dale, Mar 11 2015 *)
fQ[n_]:=PowerMod[77, n, n] == 0; Select[Range[6 10^6], fQ] (* Vincenzo Librandi, Jun 27 2016 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(11),N=11^n;while(N<=lim,listput(v,N);N*=7));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(77*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) = (7*11)/((7-1)*(11-1)) = 77/60. - Amiram Eldar, Sep 23 2020
a(n) ~ exp(sqrt(2*log(7)*log(11)*n)) / sqrt(77). - Vaclav Kotesovec, Sep 23 2020

A080682 19-smooth numbers: numbers whose prime divisors are all <= 19.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 95, 96, 98, 99, 100

Offset: 1

Cino Hilliard, Mar 02 2003

William A. Tedeschi, Table of n, a(n) for n = 1..10000

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080681, A080683.

[n: n in [1..100] | PrimeDivisors(n) subset PrimesUpTo(19)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n*17^o*19^p, {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)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}, {o, 0, Log[17, mx/(2^i*3^j*5^k*7^l*11^m*13^n)]}, {p, 0, Log[19, mx/(2^i*3^j*5^k*7^l*11^m*13^n*17^o)]}] (* Robert G. Wilson v, Jan 19 2016 *)
Select[Range[100],Max[FactorInteger[#][[All,1]]]<20&] (* Harvey P. Dale, Sep 20 2018 *)

test(n)= {m=n; forprime(p=2,19, while(m%p==0,m=m/p)); return(m==1)}
for(n=1,200,if(test(n),print1(n",")))

list(lim,p=19)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=19): # 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(agen(), 72))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log
def A080682(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 g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,19)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 19} p/(p-1) = (2*3*5*7*11*13*17*19)/(1*2*4*6*10*12*16*18) = 323323/55296. - Amiram Eldar, Sep 22 2020

cp's OEIS Frontend

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

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A266755 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^4)).

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A159991 Powers of 60: a(n) = 60^n.

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A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits, leading zeros omitted.

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

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A266755 Expansion of 1/((1-x^2)(1-x^3)(1-x^4)).