A002183 Number of divisors of n-th highly composite number.

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, 100, 108, 120, 128, 144, 160, 168, 180, 192, 200, 216, 224, 240, 256, 288, 320, 336, 360, 384, 400, 432, 448, 480, 504, 512, 576, 600, 640, 672, 720, 768, 800, 864, 896

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Record values of tau(n).

RECORDS transform of A000005.

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

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

All numbers of the form 9*2^n are present for n=0 through n=30. - Richard Peterson, Sep 07 2024

References

• S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 87.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• A. Flammenkamp, First 1200 highly composite numbers.
• Graeme McRae, Highly Composite Numbers.
• S. Ramanujan, Table of First 103 Highly Composite Numbers.
• N. J. A. Sloane, Transforms.
• Eric Weisstein's World of Mathematics, Highly Composite Number.

Crossrefs

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

Programs

Haskell

import Data.List (nub)
a002183 n = a002183_list !! (n-1)
a002183_list = nub $ map (a000005 . a061799) [1..]
-- Reinhard Zumkeller, Apr 01 2011

Mathematica

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 *)
DeleteDuplicates[DivisorSigma[0,Range[3*10^6]],GreaterEqual] (* The program generates the first 42 terms of the sequence. *) (* Harvey P. Dale, Aug 12 2025 *)

Formula

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

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

a(A261100(n)) = A070319(n). - Antti Karttunen, Jun 06 2017

a(n) = A329605(A329902(n)). - Antti Karttunen, Jan 14 2020

Extensions

More terms from Robert G. Wilson v, Jul 24 2002