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%I M0477 N0177 #125 Sep 11 2023 10:23:29
%S 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,
%T 42,45,48,49,50,54,56,60,63,64,70,72,75,80,81,84,90,96,98,100,105,108,
%U 112,120,125,126,128,135,140,144,147,150,160,162,168,175,180,189,192
%N 7-smooth numbers: positive numbers whose prime divisors are all <= 7.
%C Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.
%C Successive numbers k such that phi(210k) = 48k. - _Artur Jasinski_, Nov 05 2008
%C The divisors of 10! (A161466) are a finite subsequence. - _Reinhard Zumkeller_, Jun 10 2009
%C Numbers n such that A198487(n) > 0 and A107698(n) > 0. - _Jaroslav Krizek_, Nov 04 2011
%C A262401(a(n)) = a(n). - _Reinhard Zumkeller_, Sep 25 2015
%C Numbers which are products of single-digit numbers. - _N. J. A. Sloane_, Jul 02 2017
%C 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
%C Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - _Federico Provvedi_, Jun 06 2022
%C 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
%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A002473/b002473.txt">Table of n, a(n) for n = 1..10000</a> (first 5841 terms from N. J. A. Sloane)
%H Raphael Schumacher, <a href="https://arxiv.org/abs/1608.06928">The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers</a>, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
%H University of Ulm, <a href="http://www.informatik.uni-ulm.de/acm/Locals/1996/number.sol">The first 5842 terms</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>
%F A006530(a(n)) <= 7. - _Reinhard Zumkeller_, Apr 01 2012
%F 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
%t Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&]
%t aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* _Artur Jasinski_, Nov 05 2008 *)
%t mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* _Harvey P. Dale_, Aug 13 2012 *)
%t 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 *)
%o (PARI) test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1)
%o for(n=1,200,if(test(n),print1(n",")))
%o (PARI) is_A002473(n)=n<11||vecmax(factor(n,8)[,1])<8 \\ _M. F. Hasler_, Jan 16 2015
%o (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
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, fromList, union)
%o a002473 n = a002473_list !! (n-1)
%o a002473_list = f $ singleton 1 where
%o f s = x : f (s' `union` fromList (map (* x) [2,3,5,7]))
%o where (x, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Mar 08 2014, Apr 02 2012, Apr 01 2012
%o (Magma) [n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // _Bruno Berselli_, Sep 24 2012
%o (Python)
%o import heapq
%o from itertools import islice
%o from sympy import primerange
%o def A002473gen(p=7): # generate all p-smooth terms
%o v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
%o while True:
%o v = heapq.heappop(h)
%o if v != oldv:
%o yield v
%o oldv = v
%o for p in psmooth_primes:
%o heapq.heappush(h, v*p)
%o print(list(islice(A002473gen(), 65))) # _Michael S. Branicky_, Nov 19 2022
%Y Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.
%Y Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.
%Y Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.
%Y Cf. A262401.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 23 1999
%E Additional comments from Michel Lecomte, Jun 09 2007
%E Edited by _M. F. Hasler_, Jan 16 2015
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