A263479 Total number of n-digit positive integers with multiplicative digital root value 5.
1, 6, 33, 132, 435, 1466, 5341, 18656, 58029, 159430, 392601, 882036, 1836159, 3586506, 6638885, 11738656, 19952441, 32768742, 52220113, 81029700, 122785131, 182142906, 265066605, 379102400, 533695525, 740551526, 1014046281, 1371688948, 1834642167, 2428304010
Offset: 1
Links
- Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Programs
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Mathematica
Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &] (* Michael De Vlieger, Oct 21 2015 *)
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PARI
t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k} a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 5, 1, 0)); \\ Altug Alkan, Oct 19 2015
Formula
A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + a(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).
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
From Chai Wah Wu, Apr 17 2024: (Start)
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.
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)
Extensions
a(9)-a(30) from Hiroaki Yamanouchi, Oct 25 2015
Comments