A070319 a(n) = Max_{k=1..n} tau(k) where tau(x)=A000005(x) is the number of divisors of x.

1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset

1,2

Comments

Is this the same as A068509? - David Scambler, Sep 10 2012

They are different even asymptotically: A068509(n)=O(sqrt(n)), while a(n) does not have polynomial growth. One example where the sequences differ: a(625) = 24 < A068509(625). (The inequality is implied by the set {1,2,..,25} where each pair of the elements has lcm <= 625.) - Max Alekseyev, Sep 11 2012

The two sequences first differ when n = 336, due to the set of 21 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 21, 24, 30, 36, 42, 48} where each pair of elements has lcm <= 336, while no positive integer <= 336 has more than 20 divisors. Therefore A068509(336) = 21 and A070319(336) = 20. - William Rex Marshall, Sep 11 2012

Indices of records give A002182. - Omar E. Pol, Feb 18 2023

References

Sándor, J., Crstici, B., Mitrinović, Dragoslav S. Handbook of Number Theory I. Dordrecht: Kluwer Academic, 2006, p. 44.

S. Wigert, Sur l'ordre de grandeur du nombre des diviseurs d'un entier, Arkiv. for Math. 3 (1907), 1-9.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

Formula

a(n) = exp(log(2) log(n) / log(log(n)) + O(log(n) log(log(log(n))) / (log(log(n)))^2)). (See Sándor reference for more formulas.) - Eric M. Schmidt, Jun 30 2013

a(n) = A002183(A261100(n)). - Antti Karttunen, Jun 06 2017

Mathematica

a = {0}; Do[AppendTo[a, Max[DivisorSigma[0, n], a[[n]]]], {n, 120}]; Rest@ a (* Michael De Vlieger, Sep 29 2015 *)

Prog

(PARI) a(n)=vecmax(vector(n, k, numdiv(k)))
(PARI) v=vector(100); v[1]=1; for(n=2, #v, v[n]=max(v[n-1], numdiv(n))); v \\ Charles R Greathouse IV, Sep 12 2012
(PARI) A070319(n, m=1, s=2)={for(k=s, n, m<numdiv(k) && m=numdiv(k)); m} /* Although this should statistically require more assignments, the simple for() loop is faster than a forstep(k=n, s, -1) loop. To speed up the computation, give as 2nd and 3rd (optional) arguments earlier computed values, e.g. m=a(n-1) and s=n, cf. the example below. */  \\ M. F. Hasler, Sep 12 2012
(PARI) {a=0; for(n=1, 100, print1(a=A070319(n, a, n), ", "))} /* Using this pattern, computation of a(1..10^6) is faster than "normal" computation of a(1..3000). */
(Haskell)
a070319 n = a070319_list !! (n-1)
a070319_list = scanl1 max $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011

Crossrefs

Cf. A000005, A002182, A002183, A261100, A261104.

Sequence in context: A263089 A340611 A068509 * A057142 A320837 A098388

Adjacent sequences: A070316 A070317 A070318 * A070320 A070321 A070322

Keyword

easy,nonn

Author

Benoit Cloitre, May 11 2002

Status

approved