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A260900
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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.
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1
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2, 4, 174, 20056, 9026066, 9612284, 9612296, 9612298, 9612308, 9612310, 9612312, 9612946, 9612954, 9612962, 9612966, 9612968, 9613074, 9613394, 9667944, 10138460, 10144636, 10144638, 10144640, 10144650, 10144712, 10144756, 10144758, 10144760, 10144770
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OFFSET
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1,1
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COMMENTS
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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.
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.
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).
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:
Row # terms list of terms
==== ======= ===========================================
2 2 2, 4
3 1 174
4 1 20056
5 46 9026066, 9612284, ..., 10384656;
6 80 12436651810, 12438872740, ..., 13770404636;
7 1 45381851638748
8 1 282633399694638258
9 1 9255986333928835642154
10 1 245836727707164139860503406
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LINKS
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CROSSREFS
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KEYWORD
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nonn,tabf,base
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AUTHOR
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STATUS
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approved
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