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A263479 Total number of n-digit positive integers with multiplicative digital root value 5.

%I A263479 #39 Apr 19 2024 02:04:01
%S A263479 1,6,33,132,435,1466,5341,18656,58029,159430,392601,882036,1836159,
%T A263479 3586506,6638885,11738656,19952441,32768742,52220113,81029700,
%U A263479 122785131,182142906,265066605,379102400,533695525,740551526,1014046281,1371688948,1834642167,2428304010
%N A263479 Total number of n-digit positive integers with multiplicative digital root value 5.
%C A263479 First differences of A263473.
%H A263479 Hiroaki Yamanouchi, <a href="/A263479/b263479.txt">Table of n, a(n) for n = 1..50</a>
%H A263479 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A263479 A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + a(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).
%F A263479 a(n) = (1/720)*(3*n^8 + 6*n^7 - 664*n^6 + 6270*n^5 - 25783*n^4 + 55164*n^3 - 57796*n^2 + 23520*n). - _Sergio Pimentel_, Mar 27 2024
%F A263479 From _Chai Wah Wu_, Apr 17 2024: (Start)
%F A263479 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.
%F A263479 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)^9. (End)
%t A263479 Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &] (* _Michael De Vlieger_, Oct 21 2015 *)
%o A263479 (PARI) t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
%o A263479 a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 5, 1, 0)); \\ _Altug Alkan_, Oct 19 2015
%Y A263479 Cf. A031347, A034052, A263473.
%K A263479 nonn,base
%O A263479 1,2
%A A263479 _Martin Renner_, Oct 19 2015
%E A263479 a(9)-a(30) from _Hiroaki Yamanouchi_, Oct 25 2015