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 A051037 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5. 89
 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. 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 Also called "harmonic whole numbers", see Howard and Longair, 1982, Table I, page 121. - Hugo Pfoertner, Jul 16 2020 LINKS T. D. Noe and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from T. D. Noe M. J. Dominus, Infinite Lists in Perl. Deborah Howard, Malcolm Longair, Harmonic Proportion and Palladio's "Quattro Libri", Journal of the Society of Architectural Historians (1982) 41 (2): 116-143. Rosetta Code, A collection of computer codes to compute 5-smooth numbers. Raphael Schumacher, The Formula for the Distribution of the 3-Smooth Numbers, 5-Smooth, 7-Smooth and all other 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)=1} x^a(n) = Sum_{n>=1}-MÃ¶bius(30*n)*x^n/(1-x^n). - Paul D. Hanna, Sep 18 2011 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 MAPLE A051037 := proc(n)     option remember;     local a;     if n = 1 then         1;     else         for a from procname(n-1)+1 do             numtheory[factorset](a) minus {2, 3, 5 } ;             if % = {} then                 return a;             end if;         end do:     end if; end proc: seq(A051037(n), n=1..100) ; # R. J. Mathar, Nov 05 2017 MATHEMATICA 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. A006530, A112757, A112758, A112759, A112763, A112764, A291719. Subsequences: A003592, A003593, A257997. Sequence in context: A097752 A014866 A051661 * A250089 A257997 A070023 Adjacent sequences:  A051034 A051035 A051036 * A051038 A051039 A051040 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified October 19 20:17 EDT 2020. Contains 337892 sequences. (Running on oeis4.)