A165256 Numbers whose number of distinct prime factors equals the number of digits in the number.

2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 102, 105, 110, 114, 120, 126, 130, 132

Offset: 1

Author

Parthasarathy Nambi, Sep 11 2009

Keywords

• base
• nonn
• fini
• full

Comments

Is this sequence finite? If the answer is yes, is there any estimate for the number of terms of this sequence? - Parthasarathy Nambi, Nov 16 2009

This sequence is finite since there are only finitely many primes less than 10. - Charles R Greathouse IV, Feb 04 2013

Specifically, all terms have <= 10 digits since primorial(k) = A002110(k) has > k digits for k > 10. - Michael S. Branicky, Apr 13 2023

Examples

			The number of distinct prime factors of 4 is 1, which is the same as the number of digits in 4, so 4 is in the sequence.
The number of distinct prime factors of 21 is 2, which is the same as the number of digits in 21, so 21 is in the sequence.
However, 25 is NOT in the sequence because the number of distinct prime factors of 25 is 1, which does not match the number of digits in 25.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..7812 (complete sequence)
• Michael S. Branicky, Python program

Crossrefs

Cf. A001221, A055642, A115024, A002110.

Programs

Maple

omega := proc(n) nops(numtheory[factorset](n)) ; end: A055642 := proc(n) max(1, ilog10(n)+1) ; end: A165256 := proc(n) option remember; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A055642(a) = omega(a) then RETURN(a) ; fi; od: fi; end: seq(A165256(n),n=1..120) ; # R. J. Mathar, Sep 17 2009

Mathematica

Select[Range[200], IntegerLength[#] == Length[FactorInteger[#]] &]  (* Harvey P. Dale, Mar 20 2011 *)

PARI

is(n)=#Str(n)==omega(n) \\ Charles R Greathouse IV, Feb 04 2013

Python

# see link for alternate version producing full sequence instantly
from sympy import primefactors
def ok(n): return len(primefactors(n)) == len(str(n))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 13 2023

Extensions

Extended by R. J. Mathar, Sep 17 2009