%I #65 Sep 16 2024 14:45:27

%S 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,

%T 35,36,40,42,44,45,48,49,50,54,55,56,60,63,64,66,70,72,75,77,80,81,84,

%U 88,90,96,98,99,100,105,108,110,112,120,121,125,126,128,132,135,140

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

%C A155182 is a finite subsequence. - _Reinhard Zumkeller_, Jan 21 2009

%C From _Federico Provvedi_, Jul 09 2022: (Start)

%C 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).

%C 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)

%H David A. Corneth, <a href="/A051038/b051038.txt">Table of n, a(n) for n = 1..10000</a> (First 5000 terms from Reinhard Zumkeller)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.

%F 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

%t 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 *)

%t aQ[n_]:=Max[First/@FactorInteger[n]]<=11; Select[Range[140],aQ[#]&] (* _Jayanta Basu_, Jun 05 2013 *)

%t Block[{k=5,primorial:=Times@@Prime@Range@#&},Select[Range@200,#*EulerPhi@primorial@k==EulerPhi[#*primorial@k]&]] (* _Federico Provvedi_, Jul 09 2022 *)

%o (PARI) test(n)=m=n; forprime(p=2,11, while(m%p==0,m=m/p)); return(m==1)

%o for(n=1,200,if(test(n),print1(n",")))

%o (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

%o (Magma) [n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(11)]; // _Bruno Berselli_, Sep 24 2012

%o (Python)

%o import heapq

%o from itertools import islice

%o from sympy import primerange

%o def agen(p=11): # generate all p-smooth terms

%o v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))

%o while True:

%o v = heapq.heappop(h)

%o if v != oldv:

%o yield v

%o oldv = v

%o for p in psmooth_primes:

%o heapq.heappush(h, v*p)

%o print(list(islice(agen(), 67))) # _Michael S. Branicky_, Nov 20 2022

%o (Python)

%o from sympy import integer_log, prevprime

%o def A051038(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o 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))

%o def f(x): return n+x-g(x,11)

%o return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024

%Y Subsequence of A033620.

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

%Y A000010, A002110.

%K easy,nonn

%O 1,2

%A _Eric W. Weisstein_