1,1

A randomly selected n-digit number (uniformly distributed on 10^(n-1) to 10^n-1) has at least a 50% probability of having a prime factor at least as large as a(n).

D. E. Knuth, The Art of Computer Programming, Seminumerical Algorithms, Addison-Wesley, Reading, MA, 1969, Vol. 2.

The largest prime divisors of the non-unit 1-digit numbers are 2, 3, 2, 5, 3, 7, 2 and 3 respectively, with median 3.

Of the 90 2-digit numbers, there are 45 whose largest prime divisor is 11 or less and 45 whose largest prime divisor is 13 or greater, so any of 11, 12, or 13 could be used for the second term, although the arithmetic average of the endpoints is commonly used.

f[n_] := Block[{k = If[n == 1, 1, 0], lst = {}, pt = 10^(n - 1)}, While[k < 9*pt, AppendTo[lst, FactorInteger[pt + k][[ -1, 1]]]; k++ ]; Median@ lst]; (* Robert G. Wilson v (rgwv(at)rgwv.com), Dec 14 2006 *)

n = 1;

a = 2 | 3 | 2 | 5 | 3 | 7 | 2 | 3;

meana = meanc(a);

mediana = median(a);

format /rdn 1, 0;

print n; ; "-digit numbers:";

print " Median = "; ; mediana;

format /rdn 10, 5;

print " Mean = "; ; meana;

print;

b = 1 | a;

dim = 1;

_01: wait;

n = n+1;

dim = 10*dim;

a = b | zeros(9*dim, 1);

i = dim;

do until i == 10*dim;

if i == 2*floor(i/2);

a[i] = a[i/2];

else;

p = firstp(i);

if p == i;

a[i] = i;

else;

a[i] = a[i/p];

endif;

endif;

i = i+1;

endo;

b = a[dim:10*dim-1];

meana = meanc(b);

mediana = median(b);

format /rdn 1, 0;

print n; ; "-digit numbers:";

print " Median = "; ; mediana;

format /rdn 10, 5;

print " Mean = "; ; meana;

print;

b = a;

goto _01;

proc firstp(n);

local i;

i = 3;

do until i > sqrt(n);

if n == i*floor(n/i);

retp(i);

endif;

i = i+2;

endo;

retp(n);

endp;

Cf. A046731.

Sequence in context: A010736 A007198 A000256 this_sequence A138269 A026781 A110122

Adjacent sequences: A124199 A124200 A124201 this_sequence A124203 A124204 A124205

base,nonn

Mark Thornquist (mthornqu(AT)fhcrc.org), Dec 07 2006

Edited by Robert G. Wilson v (rgwv(at)rgwv.com), Dec 14 2006