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

1, 7, 40, 172, 607, 2073, 7414, 26070, 84099, 243529, 636130, 1518166, 3354325, 6940831, 13579716, 25318372, 45270813, 78039555, 130259668, 211289368, 334074499, 516217405, 781284010, 1160386410, 1694081935, 2434633461, 3448679742, 4820368690, 6655010857

Offset: 1

Author

Martin Renner, Oct 19 2015

Keywords

• nonn
• base

Comments

Partial sums of A263479.

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A031347, A034052, A263479.

Programs

Mathematica

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 *)

PARI

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 5, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + a(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

From Chai Wah Wu, Apr 17 2024: (Start)

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.

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)

Extensions

a(9)-a(29) from Hiroaki Yamanouchi, Oct 25 2015