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A071521 Number of 3-smooth numbers <= n. 7
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

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

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-Fran├žois 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

(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 June 28 04:43 EDT 2017. Contains 288813 sequences.