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A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5. 61
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; text; internal format)
OFFSET

1,2

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]. - Artur Jasinski, Nov 05 2008

Where record values greater than 1 occur in A165704: A165705(n)=A165704(a(n)). - Reinhard Zumkeller, Sep 26 2009

A051916 is a subsequence. - Reinhard Zumkeller, 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

A006530(a(n)) <= 5. - Reinhard Zumkeller, May 16 2015

LINKS

T. D. Noe and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from T. D. Noe

Benoit Cloitre, Plot of abs(f(n)-s(n)) vs its mean values (blue) and vs loglog(n) (red)

M. J. Dominus, Infinite Lists in Perl

Rosetta Code, A collection of computer codes to compute 5-smooth numbers

Raphael Schumacher, The Formula for the Distribution of the 3-Smooth Numbers, arXiv:1608.06928 [math.NT], 2016.

Sci.math, Ugly numbers

Eric Weisstein's World of Mathematics, Smooth Number

Wikipedia, Regular number

FORMULA

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

The characteristic function of this sequence is given by:

  Sum_{n>=1} x^a(n) = Sum_{n>=1}-möbius(30*n)*x^n/(1-x^n). - Paul D. Hanna, Sep 18 2011

MATHEMATICA

aa = {}; Do[If[8 n - EulerPhi[30 n] == 0, AppendTo[aa, n]], {n, 1, 405}]; aa (* Artur Jasinski, 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 &] (* Robert G. Wilson v *)

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)=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

(PARI) smooth(P:vec, lim)={ my(v=List([1]), nxt=vector(#P, i, 1), indx, t);

while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx);

if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++);

Vec(v)

};

smooth([2, 3, 5], 1e4) \\ Charles R Greathouse IV, Dec 03 2013

(MAGMA) [n: n in [1..500] | PrimeDivisors(n) subset [2, 3, 5]]; // Bruno Berselli, Sep 24 2012

(PARI) is_A051037(n)=n<7||vecmax(factor(n, 5)[, 1])<7 \\ M. F. Hasler, Jan 16 2015

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a051037 n = a051037_list !! (n-1)

a051037_list = f $ singleton 1 where

   f s = y : f (insert (5 * y) $ insert (3 * y) $ insert (2 * y) s')

               where (y, s') = deleteFindMin s

-- Reinhard Zumkeller, May 16 2015

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, A006530.

Subsequences: A003595, A003592, A257997.

Sequence in context: A097752 A014866 A051661 * A250089 A257997 A070023

Adjacent sequences:  A051034 A051035 A051036 * A051038 A051039 A051040

KEYWORD

easy,nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified December 7 05:39 EST 2016. Contains 278841 sequences.