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A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.
(Formerly M0477 N0177)
150
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.
Successive numbers k such that phi(210k) = 48k. - Artur Jasinski, Nov 05 2008
The divisors of 10! (A161466) are a finite subsequence. - Reinhard Zumkeller, Jun 10 2009
Numbers n such that A198487(n) > 0 and A107698(n) > 0. - Jaroslav Krizek, Nov 04 2011
A262401(a(n)) = a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers which are products of single-digit numbers. - N. J. A. Sloane, Jul 02 2017
Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - Richard Locke Peterson, May 09 2020
Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - Federico Provvedi, Jun 06 2022
The nonprimes with this property are all terms except for 2, 3, 5 and 7, i.e.: (1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, ...); the composite terms are all but the first one of this subsequence. ["Trivial" data provided mainly for search purpose.] - M. F. Hasler, Jun 06 2023
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
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
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 *)
PROG
(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", ")))
(PARI) is_A002473(n)=n<11||vecmax(factor(n, 8)[, 1])<8 \\ M. F. Hasler, Jan 16 2015
(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
(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
(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
CROSSREFS
Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.
Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.
Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.
Cf. A262401.
Sequence in context: A357315 A079333 A063938 * A174995 A161466 A178863
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Dec 23 1999
Additional comments from Michel Lecomte, Jun 09 2007
Edited by M. F. Hasler, Jan 16 2015
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)