A167050 Squarefree numbers with as many decimal digits as distinct prime factors.

2, 3, 5, 7, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310

Offset: 1

Author

Claudio Meller, Oct 27 2009

Keywords

• nonn
• base
• fini
• full

Comments

From Bernard Schott, Feb 02 2013: (Start)

These numbers appear in 1999 during the XV Gara Nazionale di Matematica, exercise 2, in Italia. [See Link]

Another definition (1): If p_1 < p_2 < p_3 < ... < p_r are r distinct primes, then n is in this sequence if 10^r <= n = p_1*p_2*...*p_r < 10^(r+1).

Another definition (2): If p_1 < p_2 < p_3 < ... < p_r are r distinct primes, then n = p_1*p_2*...*p_r has r digits in base ten.

These numbers are called "equilibrato" in Italian and translated "balanced" in English [see reference Crux Mathematicorum], I propose "nombres équilibrés" in French.

This sequence is finite, a proof without words:

2*3*5*7*11*13*17*19*23*29 = 6469693230 < 10^{10}

2*3*5*7*11*13*17*19*23*29*31 = 200560490130 > 10^{11}.

Two natural open questions:

--> 1) What is the last term of this sequence?

The last term is 9592993410 = 2*3*5*7*11*13*17*19*23*43.

--> 2) How many numbers in this sequence?

This sequence contains 4352 elements.

Concerning these two questions, I used the French mathematical forum les-mathematiques.net with the help of "JLT" and "Juge Ti" to confirm and solve them: see Link.

Subsequence of A115024. (End)

Examples

			138 = 2*3*23 and 138 is squarefree with three digits.
		

References

• R. E. Woodrow, The Olympiad Corner, No. 226, Crux Mathematicorum, v28-n8(2002), 481.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..4352 (complete sequence).
• XV Gara Nazionale di Matematica, Cesenatico, 7 Maggio 1999, exercise 2.
• Bernard Schott, List of the 4352 terms of the sequence A167050.
• Juge Ti, Bernard Schott, Jean-Louis Tu and Norbert Verdier, QDV 17: Question ouverte sans titre (French mathematical forum les-mathematiques.net).
• Index to sequences related to Olympiads and other Mathematical competitions.

Crossrefs

Intersection of A165256 and A115024.

Cf. A115024, A165256.

Programs

Maple

A001221 := proc(n) nops(numtheory[factorset](n)) ; end:
A055642 := proc(n) max(1,ilog10(n)+1) ; end:
isA167050 := proc(n) numtheory[issqrfree](n) and A055642(n) = A001221(n) end:
for n from 1 to 300 do if isA167050(n) then printf("%d,",n) ; fi; end do; # R. J. Mathar, Nov 03 2009
A Maple program is proposed by "Juge Ti" on the French mathematical forum in link for answering to the two questions (last number and cardinal of this set).

Mathematica

Select[Range[400],SquareFreeQ[#]&&PrimeNu[#]==IntegerLength[#]&] (* Harvey P. Dale, Jun 26 2021 *)

PARI

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

Formula

Intersection of A005117 and A165256.

Extensions

Definition rephrased and formula added by R. J. Mathar, Nov 05 2009