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Total number of divisors of all n-digit numbers.
2

%I #19 Jan 13 2020 12:00:19

%S 23,450,6580,86590,1073071,12803271,148755315,1694786187,19020186047,

%T 210925125565,2316483913054,25237165712764,273094922940644,

%U 2938181887791268,31454145461543161,335264720452385137,3559879862893130917,37671125212625723995,397434517963203503069

%N Total number of divisors of all n-digit numbers.

%C Partial sums of A095256.

%H Andrew Howroyd, <a href="/A174425/b174425.txt">Table of n, a(n) for n = 1..36</a>

%F From _Andrew Howroyd_, Jan 13 2020: (Start)

%F a(n) = A006218(10^n-1) - A006218(10^(n-1)-1).

%F a(n) = A057494(n) - A057494(n-1) - 2*n - 1. (End)

%e For n = 1; a(1) = 23 because tau (r) of 1-digit numbers r = 1 to 9: {1, 2, 2, 3, 2, 4, 2, 4, 3}. Sum is 23.

%o (PARI) \\ too slow for n > 20; here b(n) is A006218(n).

%o b(n)={sum(k=1, sqrtint(n), n\k)*2 - sqrtint(n)^2}

%o a(n)={b(10^n-1)-b(10^(n-1)-1)} \\ _Andrew Howroyd_, Jan 13 2020

%Y Cf. A000005, A006218, A057494, A095256.

%K nonn,base

%O 1,1

%A _Jaroslav Krizek_, Nov 28 2010

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 13 2020