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%I #32 Jan 17 2024 09:12:00
%S 1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,26,27,28,29,
%T 31,32,33,34,35,36,37,38,39,41,42,43,44,46,47,48,49,51,53,57,61,62,63,
%U 64,66,67,68,71,72,73,74,75,76,77,79,81,82,83,84,86,88,89,91,92,93,94,97,98,99,111,112,113,114,115
%N Numbers with multiplicative digital root > 0.
%C Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.
%C To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - _David A. Corneth_, Sep 27 2016
%H <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%e 25 is not in this sequence because 2*5 = 10 and 1*0 = 0.
%t Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* _Michael De Vlieger_, Sep 26 2016 *)
%o (PARI) is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ _David A. Corneth_, Sep 27 2016
%Y Cf. A009994, A047726.
%Y Cf. A031347, A034048 (complement).
%Y Cf. A028843 (a subsequence).
%Y Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
%Y Cf. A052382 (a supersequence).
%K nonn,base
%O 1,2
%A _J. Lowell_, Sep 26 2016
%E More terms from _Michael De Vlieger_, Sep 26 2016
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