A101207 -id:A101207 - cp's OEIS frontend

A190940 Number of divisors of LCM(1,2,...,n)/n.

1, 1, 2, 2, 6, 4, 12, 8, 16, 18, 48, 32, 96, 72, 64, 48, 240, 128, 480, 288, 320, 384, 960, 512, 960, 1152, 960, 1152, 3840, 3072, 7680, 3072, 6912, 7680, 6144, 6144, 18432, 15360, 13824, 12288, 36864, 23040, 73728, 49152, 49152, 61440, 147456, 73728, 147456, 122880

Offset: 1

Naohiro Nomoto, May 24 2011

nonn

Also, number of sequences of d1 = 1 < d2 < ... < dk = n for some k >= 1 that are the first k divisors of some integer (cf. A378314). - Max Alekseyev, Nov 22 2024
Also, the number of distinct values taken by lcm(a,a+b,a+b+c,...,n), where positive integers a,b,c,... run over the compositions a+b+c+...=n. - Conjectured by Ridouane Oudra, Aug 24 2019; proved by Max Alekseyev, Nov 22 2024
Proof. It is clear that n | lcm(a,a+b,...,n) | lcm(1,2,...,n). Hence, lcm(a,a+b,...,n) = d*n for some d | lcm(1,2,...,n)/n. We'll show that each such d is achievable. Suppose d*n has prime factorization p1^e1 * ... * pk^ek with p1^e1 < ... < pk^ek. It is clear that pk^ek <= n, and we can take a composition (a,b,c,...) = (p1^e1, p2^e2 - p1^e1, p3^e3 - p2^e2, ..., pk^ek - p(k-1)^e(k-1), n - pk^ek), which delivers lcm(a,a+b,a+b+c,...,n) = p1^e1 * ... * pk^ek = d*n. QED - Max Alekseyev, Nov 22 2024

			Examples: for n=3 the a(3) = 2 distinct values are 3, 6. The compositions are 3, 1+2, 2+1, and 1+1+1. The values of the lcm are lcm(3)=3, lcm(1,1+2)=3, lcm(2,2+1)=6, and lcm(1,1+1,1+1+1)=6.

First difference of A378314.

Cf. A000005, A002944, A003418, A056793, A101207.

Lpsum := proc(L) local ps,k ; ps := [op(1,L)] ; for i from 2 to nops(L) do ps := [op(ps), op(-1,ps)+op(i,L)] ; end do: ps ; end proc:
A190940 := proc(n) local lc,k,c ; lc := {} ; for k from 1 to n do for c in combinat[composition](n,k) do lc := lc union { ilcm( op(Lpsum(c))) }; end do: end do: nops(lc) ; end proc: # R. J. Mathar, Jun 02 2011

a[n_] := LCM @@@ (Accumulate /@ (Permutations /@ Rest[IntegerPartitions[n]] // Flatten[#, 1]&)) // Union // Length; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Feb 27 2014 *)

a(n) = A000005(A002944(n)).

a(12)-a(20) from R. J. Mathar, Jun 02 2011
a(21)-a(24) from Alois P. Heinz, Nov 03 2011
Edited and terms a(25) onward added by Max Alekseyev, Nov 22 2024

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

Original entry on oeis.org

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

Labos Elemer, Aug 28 2000

nonn

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

			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.

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.

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

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

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 *)

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

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

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

cp's OEIS Frontend

A190940 Number of divisors of LCM(1,2,...,n)/n.

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