A260900 - OEIS

%I #10 Aug 18 2015 08:57:40

%S 2,4,174,20056,9026066,9612284,9612296,9612298,9612308,9612310,

%T 9612312,9612946,9612954,9612962,9612966,9612968,9613074,9613394,

%U 9667944,10138460,10144636,10144638,10144640,10144650,10144712,10144756,10144758,10144760,10144770

%N Table read by rows: n-th row lists all positive integers k such that exactly half the integers in 1, 2, ..., k are pandigital in base n.

%C Numbers that are pandigital in base 2 (i.e., numbers whose digits include at least one each of 0 and 1) are 2=10_2, 4=100_2, 5=101_2, 6=110_2, 8=1000_2, etc. (i.e., all positive integers not of the form 2^j-1); exactly 2/2=1 of the first 2 positive integers and exactly 4/2=2 of the first 4 positive integers are base-2 pandigital, so 2 and 4 are in the sequence. For all k > 4, there are more base-2 pandigital numbers in 1..k than base-2 nonpandigital numbers, so there are no more terms in the n=2 row.

%C In base 3, exactly half of the integers in 1..174 are pandigital, so 174 is in the sequence. Fewer than half of the integers in 1..k are pandigital for all k < 174, and more than half of the integers in 1..k are pandigital for all k > 174, so 174 is the only term in the n=3 row.

%C The 27-digit number 245836727707164139860503406, which is a(134), is the only term in the n=10 row: in base 10, exactly half of the integers in 1..a(134) are pandigital, fewer than half of the integers in 1..k are pandigital for all k < a(134), and more than half of the integers in 1..k are pandigital for all k > a(134).

%C For each of rows 2 through 10, the number of terms and a list of those terms (abridged for rows 5 and 6) are as follows:

%C Row   # terms  list of terms

%C ====  =======  ===========================================

%C    2        2  2, 4

%C    3        1  174

%C    4        1  20056

%C    5       46  9026066, 9612284, ..., 10384656;

%C    6       80  12436651810, 12438872740, ..., 13770404636;

%C    7        1  45381851638748

%C    8        1  282633399694638258

%C    9        1  9255986333928835642154

%C   10        1  245836727707164139860503406

%H Jon E. Schoenfield, <a href="/A260900/b260900.txt">Table of n, a(n) for n = 1..134</a>

%Y Cf. A171102.

%K nonn,tabf,base

%O 1,1

%A _Jon E. Schoenfield_, Aug 04 2015, expanded Aug 07 2015