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A071521
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Number of 3-smooth numbers <= n.
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| Bruce C. Berndt and Robert A. Rankin, "Ramanujan : letters and commentary", History of Mathematics Volume 9, AMS-LMS, p. 23, p. 35
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.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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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
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MATHEMATICA
| a[n_] := Sum[ MoebiusMu[6k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 75}] (* From Jean-François Alcover, Oct 11 2011, after Benoit Cloitre *)
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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
(Haskell)
a071521 n = length $ takeWhile (<= n) a003586_list
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
| Cf. A003586.
Sequence in context: A203967 A050292 A181627 * A039733 A179510 A005374
Adjacent sequences: A071518 A071519 A071520 * A071522 A071523 A071524
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 02 2002
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