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%I #171 Nov 02 2023 07:55:20
%S 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50,54,
%T 60,64,72,75,80,81,90,96,100,108,120,125,128,135,144,150,160,162,180,
%U 192,200,216,225,240,243,250,256,270,288,300,320,324,360,375,384,400,405
%N 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.
%C Sometimes called the Hamming sequence, since Hamming asked for an efficient algorithm to generate the list, in ascending order, of all numbers of the form 2^i*3^j*5^k for i,j,k >= 0. The problem was popularized by Edsger Dijkstra.
%C Numbers k such that 8*k = EulerPhi(30*k). - _Artur Jasinski_, Nov 05 2008
%C Where record values greater than 1 occur in A165704: A165705(n) = A165704(a(n)). - _Reinhard Zumkeller_, Sep 26 2009
%C A051916 is a subsequence. - _Reinhard Zumkeller_, Mar 20 2010
%C Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. - _Hugo Pfoertner_, Jul 16 2020
%C Also called ugly numbers, although it is not clear why. - _Gus Wiseman_, May 21 2021
%C Some woody bamboo species have extraordinarily long and stable flowering intervals that belong to this sequence. The model by Veller, Nowak & Davis justifies this observation from the evolutionary point of view. - _Andrey Zabolotskiy_, Jun 27 2021
%C Also those integers k for which, for every prime p > 5, p^(4*k) - 1 == 0 (mod 240*k). - _Federico Provvedi_, May 23 2022
%C As noted in the comments to A085152, Størmer's theorem implies that the only pairs of consecutive integers that appear as consecutive terms of this sequence are (1,2), (2,3), (3,4), (4,5), (5,6), (8,9), (9,10), (15,16), (24,25), and (80,81). These all represent significant musical intervals. - _Hal M. Switkay_, Dec 05 2022
%H Reinhard Zumkeller, <a href="/A051037/b051037.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Benoit Cloitre, <a href="/A051037/a051037.png">Plot of abs(f(n)-s(n)) vs its mean values (blue) and vs loglog(n) (red)</a>.
%H M. J. Dominus, <a href="http://perl.plover.com/Stream/stream.html">Infinite Lists in Perl</a>.
%H Deborah Howard and Malcolm Longair, <a href="https://doi.org/10.2307/989675">Harmonic Proportion and Palladio's "Quattro Libri"</a>, Journal of the Society of Architectural Historians (1982) 41 (2): 116-143.
%H Vaclav Kotesovec, <a href="/A051037/a051037.jpg">Plot of a(n) / (exp((6*log(2)*log(3)*log(5)*n)^(1/3))/sqrt(30)) for n = 1..1200000</a>
%H Rosetta Code, <a href="http://rosettacode.org/wiki/Hamming_numbers">A collection of computer codes to compute 5-smooth numbers</a>.
%H Raphael Schumacher, <a href="http://arxiv.org/abs/1608.06928">The Formula for the Distribution of the 3-Smooth Numbers, 5-Smooth, 7-Smooth and all other Smooth Numbers</a>, arXiv:1608.06928 [math.NT], 2016.
%H Sci.math, <a href="https://groups.google.com/g/sci.math/c/YxwCqw6p9mk">Ugly numbers</a>.
%H Carl Veller, Martin A. Nowak and Charles C. Davis, <a href="https://doi.org/10.1111/ele.12442">Extended flowering intervals of bamboos evolved by discrete multiplication</a>, Ecology Letters, 18 (2015), 653-659.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_number">Regular number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Talk%3ARegular_number">Talk:Regular number</a>. Includes a discussion of the name.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem">Størmer's theorem</a>.
%F Let s(n) = Card(k | a(k)<n) and f(n) = log(n*sqrt(30))^3/(6*log(2)*log(3)*log(5)). Then s(n) = f(n) + O(log(n)). Conjecture: s(n)=f(n) + O(log log n). For example, s(10000000) = 768 is well approximated by f(10000000) = 769.3... (see graphic given as link). - _Benoit Cloitre_, Dec 30 2001
%F The characteristic function of this sequence is given by:
%F Sum_{n>=1} x^a(n) = Sum_{n>=1} -Möbius(30*n)*x^n/(1-x^n). - _Paul D. Hanna_, Sep 18 2011
%F a(n) = A143207(n) / 30. - _Reinhard Zumkeller_, Sep 13 2011
%F A204455(15*a(n)) = 15, and only for these numbers. - _Wolfdieter Lang_, Feb 04 2012
%F A006530(a(n)) <= 5. - _Reinhard Zumkeller_, May 16 2015
%F Sum_{n>=1} 1/a(n) = Product_{primes p <= 5} p/(p-1) = (2*3*5)/(1*2*4) = 15/4. - _Amiram Eldar_, Sep 22 2020
%e From _Gus Wiseman_, May 21 2021: (Start)
%e The sequence of terms together with their prime indices begins:
%e 1: {} 25: {3,3}
%e 2: {1} 27: {2,2,2}
%e 3: {2} 30: {1,2,3}
%e 4: {1,1} 32: {1,1,1,1,1}
%e 5: {3} 36: {1,1,2,2}
%e 6: {1,2} 40: {1,1,1,3}
%e 8: {1,1,1} 45: {2,2,3}
%e 9: {2,2} 48: {1,1,1,1,2}
%e 10: {1,3} 50: {1,3,3}
%e 12: {1,1,2} 54: {1,2,2,2}
%e 15: {2,3} 60: {1,1,2,3}
%e 16: {1,1,1,1} 64: {1,1,1,1,1,1}
%e 18: {1,2,2} 72: {1,1,1,2,2}
%e 20: {1,1,3} 75: {2,3,3}
%e 24: {1,1,1,2} 80: {1,1,1,1,3}
%e (End)
%p A051037 := proc(n)
%p option remember;
%p local a;
%p if n = 1 then
%p 1;
%p else
%p for a from procname(n-1)+1 do
%p numtheory[factorset](a) minus {2, 3,5 } ;
%p if % = {} then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc:
%p seq(A051037(n),n=1..100) ; # _R. J. Mathar_, Nov 05 2017
%t mx = 405; Sort@ Flatten@ Table[ 2^a*3^b*5^c, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}] (* Or *)
%t Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (* _Robert G. Wilson v_ *)
%t With[{nn=10},Select[Union[Times@@@Flatten[Table[Tuples[{2,3,5},n],{n,0,nn}],1]],#<=2^nn&]] (* _Harvey P. Dale_, Feb 28 2022 *)
%o (PARI) test(n)= {m=n; forprime(p=2,5, while(m%p==0,m=m/p)); return(m==1)}
%o for(n=1,500,if(test(n),print1(n",")))
%o (PARI) a(n)=local(m); if(n<1,0,n=a(n-1); until(if(m=n, forprime(p=2,5, while(m%p==0,m/=p)); m==1),n++); n)
%o (PARI) list(lim)=my(v=List(),s,t); for(i=0,logint(lim\=1,5), t=5^i; for(j=0,logint(lim\t,3), s=t*3^j; while(s<=lim, listput(v,s); s<<=1))); Set(v) \\ _Charles R Greathouse IV_, Sep 21 2011; updated Sep 19 2016
%o (PARI) smooth(P:vec,lim)={ my(v=List([1]),nxt=vector(#P,i,1),indx,t);
%o while(1, t=vecmin(vector(#P,i,v[nxt[i]]*P[i]),&indx);
%o if(t>lim,break); if(t>v[#v],listput(v,t)); nxt[indx]++);
%o Vec(v)
%o };
%o smooth([2,3,5], 1e4) \\ _Charles R Greathouse IV_, Dec 03 2013
%o (PARI) is_A051037(n)=n<7||vecmax(factor(n,6)[, 1])<7 \\ _M. F. Hasler_, Jan 16 2015
%o (Magma) [n: n in [1..500] | PrimeDivisors(n) subset [2,3,5]]; // _Bruno Berselli_, Sep 24 2012
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a051037 n = a051037_list !! (n-1)
%o a051037_list = f $ singleton 1 where
%o f s = y : f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')
%o where (y, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, May 16 2015
%o (Python)
%o def isok(n):
%o while n & 1 == 0: n >>= 1
%o while n % 3 == 0: n //= 3
%o while n % 5 == 0: n //= 5
%o return n == 1 # _Darío Clavijo_, Dec 30 2022
%Y For p-smooth numbers with other values of p, see A003586, A002473, A051038, A080197, A080681, A080682, A080683.
%Y Cf. A006530, A112757, A112758, A112759, A112763, A112764, A291719.
%Y Subsequences: A003592, A003593, A257997.
%Y The partitions with these Heinz numbers are counted by A001399.
%Y The conjugate opposite is A033942, counted by A004250.
%Y The opposite is A059485, counted by A004250.
%Y The non-3-smooth case is A080193, counted by A069905.
%Y The conjugate is A037144, counted by A001399.
%Y The complement is A279622, counted by A035300.
%Y Requiring the sum of prime indices to be even gives A344297.
%Y Cf. A000244, A002182, A002183, A035301, A056239, A112798, A261144, A344293.
%K easy,nonn
%O 1,2
%A _Eric W. Weisstein_
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