A051038 11-smooth numbers: numbers whose prime divisors are all <= 11.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140

Offset: 1

Author

Eric W. Weisstein

Keywords

• easy
• nonn

Comments

A155182 is a finite subsequence. - Reinhard Zumkeller, Jan 21 2009

From Federico Provvedi, Jul 09 2022: (Start)

In general, if p=A000040(k) is the k-th prime, with k>1, p-smooth numbers are also those positive integers m such that A000010(A002110(k))*m == A000010(A002110(k)*m).

With k=5, p = A000040(5) = 11, the primorial p# = A002110(5) = 2310, and its Euler totient is A000010(2310) = 480, so the 11-smooth numbers are also those positive integers m such that 480*m == A000010(2310*m). (End)

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (First 5000 terms from Reinhard Zumkeller)
• Eric Weisstein's World of Mathematics, Smooth Number.

Crossrefs

Subsequence of A033620.

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A080197, A080681, A080682, A080683.

A000010, A002110.

Programs

Magma

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(11)]; // Bruno Berselli, Sep 24 2012

Mathematica

mx = 150; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m, {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)]}] (* Robert G. Wilson v, Aug 17 2012 *)
aQ[n_]:=Max[First/@FactorInteger[n]]<=11; Select[Range[140],aQ[#]&] (* Jayanta Basu, Jun 05 2013 *)
Block[{k=5,primorial:=Times@@Prime@Range@#&},Select[Range@200,#*EulerPhi@primorial@k==EulerPhi[#*primorial@k]&]] (* Federico Provvedi, Jul 09 2022 *)

PARI

test(n)=m=n; forprime(p=2,11, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

PARI

list(lim,p=11)=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

Python

import heapq
from itertools import islice
from sympy import primerange
def agen(p=11): # 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(), 67))) # Michael S. Branicky, Nov 20 2022

Python

from sympy import integer_log, prevprime
def A051038(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,11)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p <= 11} p/(p-1) = (2*3*5*7*11)/(1*2*4*6*10) = 77/16. - Amiram Eldar, Sep 22 2020