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

1,2

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.

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 24.

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

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 *)

Prog

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

Crossrefs

Subsequence of A033620.

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

A000010, A002110.

Sequence in context: A033637 A084034 A084347 * A140332 A368955 A155182

Adjacent sequences: A051035 A051036 A051037 * A051039 A051040 A051041

Keyword

easy,nonn

Author

Eric W. Weisstein

Status

approved