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A071521
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Number of 3-smooth numbers <= n.
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12
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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
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OFFSET
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1,2
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COMMENTS
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A 3-smooth number is a number of the form 2^x * 3^y where x >= 0 and y >= 0.
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
a(n) = Sum_{k=1..n} (floor(6^k/k)-floor((6^k-1)/k)). - Anthony Browne, May 19 2016
a(n) = Sum_{i=0..floor(log_2(n))} (floor(log_3(n/2^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_2(n/3^i)) + 1). (End)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(Haskell)
a071521 n = length $ takeWhile (<= n) a003586_list
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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STATUS
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approved
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