A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192
Offset: 1
References
- B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5841 terms from N. J. A. Sloane)
- Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
- University of Ulm, The first 5842 terms.
- Eric Weisstein's World of Mathematics, Smooth Number.
- Wikipedia, Smooth number
Crossrefs
Programs
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Haskell
import Data.Set (singleton, deleteFindMin, fromList, union) a002473 n = a002473_list !! (n-1) a002473_list = f $ singleton 1 where f s = x : f (s' `union` fromList (map (* x) [2,3,5,7])) where (x, s') = deleteFindMin s -- Reinhard Zumkeller, Mar 08 2014, Apr 02 2012, Apr 01 2012 -
Magma
[n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // Bruno Berselli, Sep 24 2012
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Mathematica
Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&] aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* Artur Jasinski, Nov 05 2008 *) mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* Harvey P. Dale, Aug 13 2012 *) mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {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)]}] (* Robert G. Wilson v, Aug 17 2012 *) -
PARI
test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1) for(n=1,200,if(test(n),print1(n",")))
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PARI
is_A002473(n)=n<11||vecmax(factor(n,8)[,1])<8 \\ M. F. Hasler, Jan 16 2015
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PARI
list(lim)=my(v=List(),t); for(a=0,logint(lim\1,7), for(b=0,logint(lim\7^a,5), for(c=0,logint(lim\7^a\5^b,3), t=3^c*5^b*7^a; while(t<=lim, listput(v,t); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017
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Python
import heapq from itertools import islice from sympy import primerange def A002473gen(p=7): # 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(A002473gen(), 65))) # Michael S. Branicky, Nov 19 2022
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Python
from sympy import integer_log def A002473(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 f(x): c = n+x for i in range(integer_log(x,7)[0]+1): i7 = 7**i m = x//i7 for j in range(integer_log(m,5)[0]+1): j5 = 5**j r = m//j5 for k in range(integer_log(r,3)[0]+1): c -= (r//3**k).bit_length() return c return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024
Formula
A006530(a(n)) <= 7. - Reinhard Zumkeller, Apr 01 2012
Sum_{n>=1} 1/a(n) = Product_{primes p <= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - Amiram Eldar, Sep 22 2020
Extensions
More terms from James Sellers, Dec 23 1999
Additional comments from Michel Lecomte, Jun 09 2007
Edited by M. F. Hasler, Jan 16 2015
Comments