Pandigital Integers remaining Pandigital on Multiplication

Theorem

Certain pandigital integers remain pandigital when multiplying them by certain single-digit integers:

$\ds 1 \, 098 \, 765 \, 432 \times 1$

$=$

$\ds 1 \, 098 \, 765 \, 432$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 2$

$=$

$\ds 2 \, 197 \, 530 \, 864$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 3$

$=$

$\ds 3 \, 296 \, 296 \, 296$

$\ds 1 \, 098 \, 765 \, 432 \times 4$

$=$

$\ds 4 \, 395 \, 061 \, 728$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 5$

$=$

$\ds 5 \, 493 \, 827 \, 160$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 6$

$=$

$\ds 6 \, 592 \, 592 \, 592$

$\ds 1 \, 098 \, 765 \, 432 \times 7$

$=$

$\ds 7 \, 691 \, 358 \, 024$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 8$

$=$

$\ds 8 \, 790 \, 123 \, 456$

which is pandigital

$\ds 1 \, 098 \, 765 \, 432 \times 9$

$=$

$\ds 9 \, 888 \, 888 \, 888$

The sequence:

$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$

contains all pandigital integers with at least $4$ nontrivial pandigital multiples, of which:

$1098765432, 1234567890$

has $5$.

This sequence is A167476 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $123,456,789$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $123,456,789$