A346418 a(n) is the exponent of the largest power of n that divides the least common multiple of {1,2,...,n} (A003418). a(1) = 1.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jul 16 2021

Keywords

• nonn

Examples

			a(2) = 1 since A003418(2) = 2, and 2^1|A003418(2).
a(30) = 2 since A003418(30) = 2329089562800 = 30^2 * 2587877292, and 30^2|A003418(30).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Erdős, Problem 10192, The American Mathematical Monthly, Vol. 99, No. 1 (1992), p. 61; An Arithmetic Function of Modest Size, solution to problem 10192 by Richard Stong, ibid., Vol. 104, No. 1 (1997), pp. 69-70.

Crossrefs

Cf. A001221, A003418, A011776.

Programs

Mathematica

a[1] = 1; a[n_] := IntegerExponent[LCM @@ Range[n], n]; Array[a, 100]

PARI

a(n) = if (n==1, 1, valuation(lcm([1..n]), n)); \\ Michel Marcus, Jul 17 2021

Formula

a(n) <= omega(n), and a(n) < omega(n) whenever omega(n) > 1.

Max_{k=2..n} a(k) ~ log(n)/(log(log(n)) + o(1)) (Erdős, 1992).