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A070319 Max( tau(k) : k=1,2,3,...,n ) where tau(n)=A000005(n) is the number of divisors of x. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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

(Haskell)

a070319 n = a070319_list !! (n-1)

a070319_list = scanl1 max $ map a000005 [1..]

-- Reinhard Zumkeller, Apr 01 2011

(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). */

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

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Last modified December 4 15:36 EST 2021. Contains 349526 sequences. (Running on oeis4.)