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A299690 Numbers without digit 1 whose multiplicative digital root is not 0.

%I A299690 #57 Nov 14 2018 14:29:39
%S A299690 2,3,4,5,6,7,8,9,22,23,24,26,27,28,29,32,33,34,35,36,37,38,39,42,43,
%T A299690 44,46,47,48,49,53,57,62,63,64,66,67,68,72,73,74,75,76,77,79,82,83,84,
%U A299690 86,88,89,92,93,94,97,98,99,222,223,224,226,227,228,229,232
%N A299690 Numbers without digit 1 whose multiplicative digital root is not 0.
%C A299690 Is this sequence infinite?
%C A299690 There are no members of this sequence with 45 to 2000 decimal digits. Perhaps the last term is a(614640917006263790) = 77333222222222222222222222222222222222222222. - _Charles R Greathouse IV_, Feb 26 2018
%C A299690 This sequence is finite. Proof: Let k be the smallest term with more than 2000 decimal digits. Then the product of decimal digits pk of k has fewer than 2001 decimal digits (otherwise k isn't the smallest term with more than 2000 decimal digits). This number pk has at least as many decimal digits as 2^2001 has, which are 603. But then it doesn't have a nonzero multiplicative digital root per the computations of _Charles R Greathouse IV_. QED. - _David A. Corneth_, Aug 23 2018
%H A299690 Charles R Greathouse IV, <a href="/A299690/b299690.txt">Table of n, a(n) for n = 1..10000</a>
%F A299690 { A052383 } intersect { A277061 }.
%e A299690 5 times 4 = 20 and 2 times 0 = 0, so 54 is not in this sequence.
%t A299690 multDigRoot[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Select[Range[500], DigitCount[#, 10, 1] == 0 && multDigRoot[#] != 0 &] (* _Alonso del Arte_, Feb 19 2018, based on _Robert G. Wilson v_'s program for A031347 *)
%o A299690 (PARI) mdr(n)=while(n>9,n=factorback(digits(n)));n
%o A299690 do(n)=my(v=List());forvec(u=vector(n,i,[2,9]), if(mdr(factorback(u)), listput(v, fromdigits(u)))); Vec(v) \\ Gives n-digit elements
%o A299690 \\ _Charles R Greathouse IV_, Feb 19 2018
%Y A299690 Cf. A031347, A052383, A277061.
%K A299690 nonn,fini,base
%O A299690 1,1
%A A299690 _J. Lowell_, Feb 19 2018