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

%I A167050 #46 Jun 26 2021 17:12:24
%S A167050 2,3,5,7,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,
%T A167050 77,82,85,86,87,91,93,94,95,102,105,110,114,130,138,154,165,170,174,
%U A167050 182,186,190,195,222,230,231,238,246,255,258,266,273,282,285,286,290,310
%N A167050 Squarefree numbers with as many decimal digits as distinct prime factors.
%C A167050 From _Bernard Schott_, Feb 02 2013: (Start)
%C A167050 These numbers appear in 1999 during the XV Gara Nazionale di Matematica, exercise 2, in Italia. [See Link]
%C A167050 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).
%C A167050 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.
%C A167050 These numbers are called "equilibrato" in Italian and translated "balanced" in English [see reference Crux Mathematicorum], I propose "nombres équilibrés" in French.
%C A167050 This sequence is finite, a proof without words:
%C A167050 2*3*5*7*11*13*17*19*23*29 = 6469693230 < 10^{10}
%C A167050 2*3*5*7*11*13*17*19*23*29*31 = 200560490130 > 10^{11}.
%C A167050 Two natural open questions:
%C A167050 --> 1) What is the last term of this sequence?
%C A167050 The last term is 9592993410 = 2*3*5*7*11*13*17*19*23*43.
%C A167050 --> 2) How many numbers in this sequence?
%C A167050 This sequence contains 4352 elements.
%C A167050 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.
%C A167050 Subsequence of A115024. (End)
%D A167050 R. E. Woodrow, The Olympiad Corner, No. 226, Crux Mathematicorum, v28-n8(2002), 481.
%H A167050 Charles R Greathouse IV, <a href="/A167050/b167050.txt">Table of n, a(n) for n = 1..4352</a> (complete sequence).
%H A167050 XV Gara Nazionale di Matematica, <a href="http://andfog.altervista.org/math/Cesenatico/Cesenatico%201999.pdf">Cesenatico, 7 Maggio 1999, exercise 2</a>.
%H A167050 Bernard Schott, <a href="/A167050/a167050.pdf">List of the 4352 terms of the sequence A167050</a>.
%H A167050 Juge Ti, Bernard Schott, Jean-Louis Tu and Norbert Verdier, <a href="http://www.les-mathematiques.net/phorum/read.php?17,813305,813305#msg-813305">QDV 17: Question ouverte sans titre</a> (French mathematical forum les-mathematiques.net).
%H A167050 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A167050 Intersection of A005117 and A165256.
%e A167050 138 = 2*3*23 and 138 is squarefree with three digits.
%p A167050 A001221 := proc(n) nops(numtheory[factorset](n)) ; end:
%p A167050 A055642 := proc(n) max(1,ilog10(n)+1) ; end:
%p A167050 isA167050 := proc(n) numtheory[issqrfree](n) and A055642(n) = A001221(n) end:
%p A167050 for n from 1 to 300 do if isA167050(n) then printf("%d,",n) ; fi; end do; # _R. J. Mathar_, Nov 03 2009
%p A167050 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).
%t A167050 Select[Range[400],SquareFreeQ[#]&&PrimeNu[#]==IntegerLength[#]&] (* _Harvey P. Dale_, Jun 26 2021 *)
%o A167050 (PARI) is(n)=issquarefree(n)&&#Str(n)==omega(n) \\ _Charles R Greathouse IV_, Feb 04 2013
%Y A167050 Intersection of A165256 and A115024.
%Y A167050 Cf. A115024, A165256.
%K A167050 nonn,base,fini,full
%O A167050 1,1
%A A167050 _Claudio Meller_, Oct 27 2009
%E A167050 Definition rephrased and formula added by _R. J. Mathar_, Nov 05 2009