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

%I A051038 #66 Feb 16 2025 08:32:41
%S A051038 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 A051038 35,36,40,42,44,45,48,49,50,54,55,56,60,63,64,66,70,72,75,77,80,81,84,
%U A051038 88,90,96,98,99,100,105,108,110,112,120,121,125,126,128,132,135,140
%N A051038 11-smooth numbers: numbers whose prime divisors are all <= 11.
%C A051038 A155182 is a finite subsequence. - _Reinhard Zumkeller_, Jan 21 2009
%C A051038 From _Federico Provvedi_, Jul 09 2022: (Start)
%C A051038 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 A051038 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 A051038 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 A051038 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%F A051038 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 A051038 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 A051038 aQ[n_]:=Max[First/@FactorInteger[n]]<=11; Select[Range[140],aQ[#]&] (* _Jayanta Basu_, Jun 05 2013 *)
%t A051038 Block[{k=5,primorial:=Times@@Prime@Range@#&},Select[Range@200,#*EulerPhi@primorial@k==EulerPhi[#*primorial@k]&]] (* _Federico Provvedi_, Jul 09 2022 *)
%o A051038 (PARI) test(n)=m=n; forprime(p=2,11, while(m%p==0,m=m/p)); return(m==1)
%o A051038 for(n=1,200,if(test(n),print1(n",")))
%o A051038 (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 A051038 (Magma) [n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(11)]; // _Bruno Berselli_, Sep 24 2012
%o A051038 (Python)
%o A051038 import heapq
%o A051038 from itertools import islice
%o A051038 from sympy import primerange
%o A051038 def agen(p=11): # generate all p-smooth terms
%o A051038     v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
%o A051038     while True:
%o A051038         v = heapq.heappop(h)
%o A051038         if v != oldv:
%o A051038             yield v
%o A051038             oldv = v
%o A051038             for p in psmooth_primes:
%o A051038                 heapq.heappush(h, v*p)
%o A051038 print(list(islice(agen(), 67))) # _Michael S. Branicky_, Nov 20 2022
%o A051038 (Python)
%o A051038 from sympy import integer_log, prevprime
%o A051038 def A051038(n):
%o A051038     def bisection(f,kmin=0,kmax=1):
%o A051038         while f(kmax) > kmax: kmax <<= 1
%o A051038         while kmax-kmin > 1:
%o A051038             kmid = kmax+kmin>>1
%o A051038             if f(kmid) <= kmid:
%o A051038                 kmax = kmid
%o A051038             else:
%o A051038                 kmin = kmid
%o A051038         return kmax
%o A051038     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 A051038     def f(x): return n+x-g(x,11)
%o A051038     return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024
%Y A051038 Subsequence of A033620.
%Y A051038 For p-smooth numbers with other values of p, see A003586, A051037, A002473, A080197, A080681, A080682, A080683.
%Y A051038 A000010, A002110.
%K A051038 easy,nonn
%O A051038 1,2
%A A051038 _Eric W. Weisstein_