A104659 Number of distinct prime divisors of 44...441 (with n 4s).

1, 2, 1, 2, 2, 2, 2, 4, 2, 1, 5, 3, 2, 6, 3, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 5, 1, 2, 6, 4, 4, 6, 4, 4, 4, 5, 4, 8, 4, 4, 7, 3, 2, 7, 3, 7, 4, 6, 3, 4, 6, 2, 6, 1, 4, 7, 2, 5, 4, 4, 4, 6, 4, 2, 3, 6, 3, 5, 4, 3, 11, 5, 4, 4, 5, 7, 3, 4, 3, 5, 4, 4, 3, 3, 6, 8, 3, 4, 4, 2, 6, 6, 1, 7, 8, 4, 4, 7, 4, 6, 6, 4, 4, 5, 6

Offset: 1

Author

Parthasarathy Nambi, Apr 21 2005, extended Aug 08 2010

Keywords

• nonn
• base

Comments

There are very few primes in this sequence. 41 appears as the smallest prime divisor frequently. There are many semiprimes.

41 is prime.

4441 is prime.

44444 444441 is prime.

4444 444444 444444 444444 444441 is prime.

4444444444444444444444444444444444444444444444444444441 is prime.

Computed using www.alpertron.com.ar/ECM.HTM

Examples

			The number of distinct prime divisors of 441 is 2.
The number of distinct prime divisors of 44444444444444444444444444444441 is four.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..199
• Dario Alpern, Factorization using the Elliptic Curve Method
• Makoto Kamada, Factorizations of 44...441.

Crossrefs

Cf. A001221, A104564, A104517, A104484, A173768.

Programs

Mathematica

f[n_] := Length@ FactorInteger[(4*10^(n + 1) - 31)/9]; Array[f, 105] (* Robert G. Wilson v, Aug 09 2010 *)
PrimeNu/@Rest[FromDigits/@Table[PadLeft[{1},n,4],{n,110}]] (* Harvey P. Dale, Mar 16 2012 *)

Formula

a(n) = A001221(A173768(n+1)). - Amiram Eldar, Jan 24 2020

Extensions

a(32) - a(105) from Robert G. Wilson v, Aug 09 2010