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A051037
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5-smooth numbers: i.e. numbers whose prime divisors are all <= 5.
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45
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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, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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.
Successive numbers k such that 8 k = EulerPhi[30 k]. [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]
Where record values greater than 1 occur in A165704: A165705(n)=A165704(a(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 26 2009]
A051916 is a subsequence. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 20 2010]
a(n) = A143207(n) / 30. [Reinhard Zumkeller, Sep 13 2011]
A204455(15*a(n)) = 15, and only for these numbers. [Wolfdieter Lang, Feb 04 2012]
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..1000
M. J. Dominus, Infinite Lists in Perl.
Sci.math, Ugly numbers
Eric Weisstein's World of Mathematics, Smooth Number
Wikipedia, Regular number [From Artur Jasinski (grafix(AT)csl.pl), Nov 06 2008]
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FORMULA
| Let s(n)=Card(k | a(k)<n) and f(n) = ln(n*sqrt(30))^3/(6*ln(2)*ln(3)*ln(5)). Then s(n) = f(n) + O(lnln(n)). For example s(10000000)=768 is well approximated by f(10000000)=769, 3... - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 30 2001
The characteristic function of this sequence is given by:
Sum_{n>=1} x^a(n) = Sum_{n>=1} -moebius(30*n)*x^n/(1-x^n). [Paul D. Hanna, Sep 18 2011]
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MATHEMATICA
| aa = {}; Do[If[8 n - EulerPhi[30 n] == 0, AppendTo[aa, n]], {n, 1, 405}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]
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 *)
Select[ Range@ 405, Last@ Map[First, FactorInteger@ #] < 7 &] (* RGWv *)
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PROG
| (PARI) test(n)= {m=n; forprime(p=2, 5, while(m%p==0, m=m/p)); return(m==1)} for(n=1, 500, if(test(n), print1(n", ")))
(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)
(PARI) list(lim)={
lim\=1;
my(v=List(), s, t);
for(i=0, log(lim+.5)\log(5),
t=5^i;
for(j=0, log(lim\t+.5)\log(3),
s=t*3^j;
while(s <= lim,
listput(v, s);
s <<= 1;
)
)
);
vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Sep 21 2011
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CROSSREFS
| For p-smooth numbers with other values of p, see A003586, A002473, A051038, A080197, A080681, A080682, A080683.
Cf. A112757, A112758, A112759, A112763, A112764, A003593.
Sequence in context: A097752 A014866 A051661 * A070023 A035303 A018609
Adjacent sequences: A051034 A051035 A051036 * A051038 A051039 A051040
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KEYWORD
| easy,nonn
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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