A385863 a(n) is the largest number of distinct prime factors a number with at most n digits can have.

2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 39, 40

Offset: 1

Author

David A. Corneth, Aug 20 2025

Keywords

• nonn
• easy
• base
• new

Comments

Also the largest k such that primorial(k) < 10^n.

"at most" in name could also be "exactly" and it gives the same data.

a(n) is the number of distinct prime factors of A091800(n).

Examples

			a(5) = 6 as primorial(6) = 30030 < 10^5 < 510510 = primorial(6 + 1) = primorial(7).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A002110, A002283, A067175, A091800, A387173.

Programs

Mathematica

Table[First[FirstPosition[#, ?(# > n &)]] - 1, {n, Last[#] - 1}] & [IntegerLength[FoldList[Times, Prime[Range[50]]]]] (* _Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = my(ulim=10^n-1, pp=1, t=0); forprime(p=2, oo, pp*=p; if(pp > ulim, return(t)); t++)

Formula

a(n) = A001221(A091800(n)).