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 A002183 Number of divisors of n-th highly composite number. (Formerly M0546 N0196) 62

%I M0546 N0196

%S 1,2,3,4,6,8,9,10,12,16,18,20,24,30,32,36,40,48,60,64,72,80,84,90,96,

%T 100,108,120,128,144,160,168,180,192,200,216,224,240,256,288,320,336,

%U 360,384,400,432,448,480,504,512,576,600,640,672,720,768,800,864,896

%N Number of divisors of n-th highly composite number.

%C Record values of tau(n).

%C RECORDS transform of A000005.

%C All powers of 2 are present through 2^17. No power of 2 above that is present at least through 2^51. - Comment from _Robert G. Wilson v_, modified by _Ray Chandler_, Nov 10 2005

%C No power of 2 above 2^17 is contained in this sequence - see McRae link for proof. - _Graeme McRae_, Apr 27 2006

%D S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 87.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ray Chandler, <a href="/A002183/b002183.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">First 1200 highly composite numbers</a>.

%H Graeme McRae, <a href="https://web.archive.org/web/20190223125015/http://2000clicks.com/mathhelp/NumberFactorsHighlyComposite.aspx">Highly Composite Numbers</a>.

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page10.htm">Table of First 103 Highly Composite Numbers</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>.

%F a(n) = A000005(A002182(n)).

%F Also record values of differences A006218(p)-A006218(p-1). These record values occur for any p = A002182(q) where q>=2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007

%F a(A261100(n)) = A070319(n). - _Antti Karttunen_, Jun 06 2017

%F a(n) = A329605(A329902(n)). - _Antti Karttunen_, Jan 14 2020

%t Reap[ For[ record = 0; n = 1, n <= 10^9, n = If[n < 60, n+1, n+60], tau = DivisorSigma[0, n]; If[tau > record, record = tau; Print[tau]; Sow[tau]]]][[2, 1]] (* _Jean-François Alcover_, Aug 13 2013 *)

%o import Data.List (nub)

%o a002183 n = a002183_list !! (n-1)

%o a002183_list = nub \$ map (a000005 . a061799) [1..]

%o -- _Reinhard Zumkeller_, Apr 01 2011

%Y Cf. A000005, A002182, A002201, A006218, A061799, A070319, A243220, A261100, A329605, A329902.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, Jul 24 2002

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Last modified August 4 01:20 EDT 2020. Contains 336201 sequences. (Running on oeis4.)