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

%I A080682 #39 Sep 16 2024 20:35:49
%S A080682 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,
%T A080682 28,30,32,33,34,35,36,38,39,40,42,44,45,48,49,50,51,52,54,55,56,57,60,
%U A080682 63,64,65,66,68,70,72,75,76,77,78,80,81,84,85,88,90,91,95,96,98,99,100
%N A080682 19-smooth numbers: numbers whose prime divisors are all <= 19.
%H A080682 William A. Tedeschi, <a href="/A080682/b080682.txt">Table of n, a(n) for n = 1..10000</a>
%F A080682 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
%t A080682 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 *)
%t A080682 Select[Range[100],Max[FactorInteger[#][[All,1]]]<20&] (* _Harvey P. Dale_, Sep 20 2018 *)
%o A080682 (PARI) test(n)= {m=n; forprime(p=2,19, while(m%p==0,m=m/p)); return(m==1)}
%o A080682 for(n=1,200,if(test(n),print1(n",")))
%o A080682 (PARI) 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
%o A080682 (Magma) [n: n in [1..100] | PrimeDivisors(n) subset PrimesUpTo(19)]; // _Bruno Berselli_, Sep 24 2012
%o A080682 (Python)
%o A080682 import heapq
%o A080682 from itertools import islice
%o A080682 from sympy import primerange
%o A080682 def agen(p=19): # generate all p-smooth terms
%o A080682     v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
%o A080682     while True:
%o A080682         v = heapq.heappop(h)
%o A080682         if v != oldv:
%o A080682             yield v
%o A080682             oldv = v
%o A080682             for p in psmooth_primes:
%o A080682                 heapq.heappush(h, v*p)
%o A080682 print(list(islice(agen(), 72))) # _Michael S. Branicky_, Nov 20 2022
%o A080682 (Python)
%o A080682 from sympy import integer_log
%o A080682 def A080682(n):
%o A080682     def bisection(f,kmin=0,kmax=1):
%o A080682         while f(kmax) > kmax: kmax <<= 1
%o A080682         while kmax-kmin > 1:
%o A080682             kmid = kmax+kmin>>1
%o A080682             if f(kmid) <= kmid:
%o A080682                 kmax = kmid
%o A080682             else:
%o A080682                 kmin = kmid
%o A080682         return kmax
%o A080682     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 A080682     def f(x): return n+x-g(x,19)
%o A080682     return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024
%Y A080682 For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080681, A080683.
%K A080682 easy,nonn
%O A080682 1,2
%A A080682 _Cino Hilliard_, Mar 02 2003