I find the pattern in the first row of your table very interesting. Is there any intuitive reason that pandigital numbers in odd bases only have very small powers of two as divisors? On Tue, Mar 31, 2020 at 7:56 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Hans Havermann <gladhobo@bell.net> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
What do 3076521984, 3718250496, and 6398410752, and no other numbers, have in common?"
They are obviously pandigital numbers and thus share the divisor 3^2. The GCD of the three numbers divided by 9 is a large power of 2. A quick check shows that no other pandigital has that large a power-of-two divisor (the next largest being for 9805234176).
Correct. My three numbers are all divisible by 2^21. No 10-digit base-10 pandigital numbers are divisible by 2^22.
I've looked into the highest powers of small primes among the n-digit base-n pandigital numbers for small n. Here's a table. As always with tables, it should be viewed in a fixed-width font.
base: 2 3 4 5 6 7 8 9 10 11 12 13
2: 1 0 3 1 11 0 5 2 21 0 29 1 3: 0 1 3 5 7 9 10 3 15 17 18 21 5: 0 1 2 1 5 5 6 9 9 11 12 13 7: 0 1 2 2 3 1 7 6 8 9 9 13 11: 0 1 1 2 2 3 5 7 5 1 8 9 13: 0 0 1 2 2 3 3 5 5 6 8 1
I can easily push it to higher primes, but not to higher bases. Computing the entries for base n will take roughly n times longer than computing them for all smaller bases put together.
I may extend this table to powers of composite numbers.
It should be obvious why the entry equals 1 whenever the row and column numbers are equal.
The "opposite" problem, finding n-digit base-n pandigital primes, isn't very interesting. There are only three of them. Can you find them? (I haven't looked into non-standard bases, such as negabinary, Fibonacci, or balanced ternary.)
Which of these rows and columns would be of interest to OEIS? None of them appear to be there yet.
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