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 A071521 Number of 3-smooth numbers <= n. 10
 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A 3-smooth number is a number of the form 2^x * 3^y where x >= 0 and y >= 0. REFERENCES Bruce C. Berndt and Robert A. Rankin, "Ramanujan : letters and commentary", History of Mathematics Volume 9, AMS-LMS, p. 23, p. 35. G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Pub., 1999, pages 67-82. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d. M. Haussman and H. N. Shapiro, On Ramanujan right triangle conjecture, Comm. Pure Appl. Math. 42 (1989), 885-889. A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 77-98; 250-251. FORMULA a(n) = Card{ k | A003586(k) <= n } Asymptotically: let a=1/(2*log(2)*log(3)), b=sqrt(6), then from Ramanujan a(n) ~ a*log(2*n)*log(3*n) or equivalently a(n) ~ a*log(b*n)^2. A022331(n) = a(A000079(n)); A022330(n) = a(A000244(n)). - Reinhard Zumkeller, May 09 2006 a(n) = Sum_{k=1..n} mu(6k)*floor(n/k). - Benoit Cloitre, Jun 14 2007 a(n) = Sum_{k=1..n} (floor(6^k/k)-floor((6^k-1)/k)). - Anthony Browne, May 19 2016 From Ridouane Oudra, Jul 17 2020: (Start) a(n) = floor(log_2(n)) + 1 + Sum_{i=0..floor(log_2(n))} floor(log_3(n/2^i)). a(n) = floor(log_3(n)) + 1 + Sum_{i=0..floor(log_3(n))} floor(log_2(n/3^i)). (End) MAPLE N:= 10000: # to get a(1) to a(N) V:= Vector(N): for y from 0 to floor(log[3](N)) do   for x from 0 to ilog2(N/3^y) do     V[2^x*3^y]:= 1 od od: convert(map(round, Statistics:-CumulativeSum(V)), list); # Robert Israel, Dec 16 2014 MATHEMATICA a[n_] := Sum[ MoebiusMu[6k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Oct 11 2011, after Benoit Cloitre *) f[n_] := Sum[Floor@Log[3, n/2^i] + 1, {i, 0, Log[2, n]}]; Array[f, 75] (* faster, or *) f[n_] := Sum[Floor@Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Array[f, 75] (* Robert G. Wilson v, Aug 18 2012 *) Accumulate[Table[If[Max[FactorInteger[n][[All, 1]]]<4, 1, 0], {n, 80}]] (* Harvey P. Dale, Jan 11 2017 *) PROG (PARI) for(n=1, 100, print1(sum(k=1, n, if(sum(i=3, n, if(k%prime(i), 0, 1)), 0, 1)), ", ")) (PARI) a(n)=sum(k=1, n, moebius(2*3*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007 (PARI) a(n)=my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1) \\ Charles R Greathouse IV, Jan 08 2018 (Haskell) a071521 n = length \$ takeWhile (<= n) a003586_list -- Reinhard Zumkeller, Aug 14 2011 CROSSREFS Cf. A003586. Sequence in context: A267530 A050292 A181627 * A204330 A225553 A039733 Adjacent sequences:  A071518 A071519 A071520 * A071522 A071523 A071524 KEYWORD easy,nice,nonn AUTHOR Benoit Cloitre, Jun 02 2002 STATUS approved

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Last modified September 18 20:58 EDT 2021. Contains 347536 sequences. (Running on oeis4.)