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A263473 Total number of positive integers < 10^n with multiplicative digital root value 5.

%I A263473 #27 Apr 19 2024 02:00:08
%S A263473 1,7,40,172,607,2073,7414,26070,84099,243529,636130,1518166,3354325,
%T A263473 6940831,13579716,25318372,45270813,78039555,130259668,211289368,
%U A263473 334074499,516217405,781284010,1160386410,1694081935,2434633461,3448679742,4820368690,6655010857
%N A263473 Total number of positive integers < 10^n with multiplicative digital root value 5.
%C A263473 Partial sums of A263479.
%H A263473 Hiroaki Yamanouchi, <a href="/A263473/b263473.txt">Table of n, a(n) for n = 1..50</a>
%H A263473 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A263473 A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + a(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).
%F A263473 From _Chai Wah Wu_, Apr 17 2024: (Start)
%F A263473 a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 10.
%F A263473 G.f.: x*(-235*x^7 + 205*x^6 + 161*x^5 + 57*x^4 - 33*x^3 + 15*x^2 - 3*x + 1)/(x - 1)^10. (End)
%t A263473 lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* _Michael De Vlieger_, Oct 21 2015 *)
%o A263473 (PARI) t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
%o A263473 a(n) = sum(i=1, 10^n - 1, if(t(i) == 5, 1, 0)); \\ _Altug Alkan_, Oct 19 2015
%Y A263473 Cf. A031347, A034052, A263479.
%K A263473 nonn,base
%O A263473 1,2
%A A263473 _Martin Renner_, Oct 19 2015
%E A263473 a(9)-a(29) from _Hiroaki Yamanouchi_, Oct 25 2015