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A080681 17-smooth numbers: numbers whose prime divisors are all <= 17.

%I A080681 #36 Sep 16 2024 18:24:16
%S A080681 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,24,25,26,27,28,
%T A080681 30,32,33,34,35,36,39,40,42,44,45,48,49,50,51,52,54,55,56,60,63,64,65,
%U A080681 66,68,70,72,75,77,78,80,81,84,85,88,90,91,96,98,99,100,102,104,105
%N A080681 17-smooth numbers: numbers whose prime divisors are all <= 17.
%H A080681 William A. Tedeschi, <a href="/A080681/b080681.txt">Table of n, a(n) for n = 1..10000</a>
%F A080681 Sum_{n>=1} 1/a(n) = Product_{primes p <= 17} p/(p-1) = (2*3*5*7*11*13*17)/(1*2*4*6*10*12*16) = 17017/3072. - _Amiram Eldar_, Sep 22 2020
%t A080681 mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n*17^o, {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)]}] (* _Robert G. Wilson v_, Aug 17 2012 *)
%o A080681 (PARI) test(n)= {m=n; forprime(p=2,17, while(m%p==0,m=m/p)); return(m==1)}
%o A080681 for(n=1,200,if(test(n),print1(n",")))
%o A080681 (PARI) list(lim,p=17)=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 A080681 (Magma) [n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(17)]; // _Bruno Berselli_, Sep 24 2012
%o A080681 (Python)
%o A080681 import heapq
%o A080681 from itertools import islice
%o A080681 from sympy import primerange
%o A080681 def agen(p=17): # generate all p-smooth terms
%o A080681     v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
%o A080681     while True:
%o A080681         v = heapq.heappop(h)
%o A080681         if v != oldv:
%o A080681             yield v
%o A080681             oldv = v
%o A080681             for p in psmooth_primes:
%o A080681                 heapq.heappush(h, v*p)
%o A080681 print(list(islice(agen(), 70))) # _Michael S. Branicky_, Nov 20 2022
%o A080681 (Python)
%o A080681 from sympy import integer_log, prevprime
%o A080681 def A080681(n):
%o A080681     def bisection(f,kmin=0,kmax=1):
%o A080681         while f(kmax) > kmax: kmax <<= 1
%o A080681         while kmax-kmin > 1:
%o A080681             kmid = kmax+kmin>>1
%o A080681             if f(kmid) <= kmid:
%o A080681                 kmax = kmid
%o A080681             else:
%o A080681                 kmin = kmid
%o A080681         return kmax
%o A080681     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 A080681     def f(x): return n+x-g(x,17)
%o A080681     return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024
%Y A080681 For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080682, A080683.
%K A080681 easy,nonn
%O A080681 1,2
%A A080681 _Cino Hilliard_, Mar 02 2003