A056793 Number of divisors of lcm(1..n).

1, 2, 4, 6, 12, 12, 24, 32, 48, 48, 96, 96, 192, 192, 192, 240, 480, 480, 960, 960, 960, 960, 1920, 1920, 2880, 2880, 3840, 3840, 7680, 7680, 15360, 18432, 18432, 18432, 18432, 18432, 36864, 36864, 36864, 36864, 73728, 73728, 147456, 147456, 147456, 147456

Offset

1,2

Comments

The ratio a(n)/a(n-1) equals 1 if n is a member of A024619, equals 2 if n is prime, and is a noninteger value if n is in A025475. The noninteger ratio never seems to exceed 3/2, but appears to equal 3/2 if n is a member of A001248. The noninteger ratio conforms to the formula 1/(1 - 1/n), which has 1 for limit and only 2 as single integer solution. In terms of coordinates (x,y), the lower values are (1/(1-1/n), 2^(n-1)) for n > 2. - Eric Desbiaux, Jul 28 2013

Conjectured partial sums of A101207. - Sean A. Irvine, Jun 25 2022

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Angad Singh, Note 106.01, The number of divisors of the LCM of the first n natural numbers, The Mathematical Gazette, Vol. 106, No. 565 (2022), pp. 116-117.

Formula

a(n) = d(lcm(1..n)) = A000005(A003418(n)).

a(n) = Product_{prime p <= n} (floor(log(n)/log(p)) + 1). - Wei Zhou, Jun 25 2011

a(n) = Product_{k>=1} (1+1/k)^pi(n^(1/k)), where pi(n) = A000720(n) (Singh, 2022). - Amiram Eldar, Aug 19 2023

Example

n = 20: lcm(1..20) = 2*2*2*2*3*3*5*7*11*13*17*19 = 232792560 and d(232792560) = 5*3*64 = 960.

Maple

A056793 := proc(n)
    numtheory[tau](lcm($1..n)) ;
end proc; # Nathaniel Johnston, Jun 26 2011

Mathematica

Table[DivisorSigma[0, LCM @@ Range[n]], {n, 50}]
Table[Product[Floor[Log[Prime[i], n]] + 1, {i, PrimePi[n]}], {n, 100}] (* Wei Zhou, Jun 25 2011 *)

Prog

(PARI) a(n)=n+=.5; prod(e=1, log(n)\log(2), (1+1/e)^primepi(n^(1/e))) \\ Charles R Greathouse IV, Jun 06 2013
(Python)
from math import lcm
from sympy import divisor_count
from itertools import accumulate, count, islice
def agen(): yield from map(divisor_count, accumulate(count(1), lcm))
print(list(islice(agen(), 46))) # Michael S. Branicky, Jun 25 2022

Crossrefs

Cf. A000005, A000720, A003418, A024619, A101207.

Sequence in context: A062857 A061799 A076868 * A137387 A137394 A062856

Adjacent sequences: A056790 A056791 A056792 * A056794 A056795 A056796

Keyword

nonn

Author

Labos Elemer, Aug 28 2000

Status

approved