Pandigital Numbers Formed From Squares

As the natural numbers become larger, it's more and more likely that they will contain all of the digits from 1 to 9 at least once. This is why the sequence of numbers containing all the digits from 1 to 9 is said to have an asymptotic density of 1. This is OEIS A294661:

A294661

Numbers whose square contains all of the digits 1 through 9.

The sequence begins:

11826, 12363, 12543, 14676, 15681, 15963, 18072, 19023, 19377, 19569, 19629, 20316, 22887, 23019, 23178, 23439, 24237, 24276, 24441, 24807, 25059, 25572, 25941, 26409, 26733, 27129, 27273, 29034, 29106, 30384, 32043, 32286, 33144, 34273, 35172, 35337, 35713, 35756, 35757, 35772, 35846, 35853, ...

I've marked the first thirty members of this sequence in blue because these numbers constitute OEIS A071519:

A071519

Numbers whose square is a zeroless pandigital number (i.e., use the digits 1 through 9 once).

Beyond 30384 (the 30th and last member of OEIS A071519), some of the digits occur more than once or a zero appears. For example: $$32043^2=1026753849 \text{ and a zero appears}$$There are 362,880 ways of arranging of the digits from 1 to 9 (factorial 9 or 9!). However, only 30 of the resulting numbers are perfect squares and these are listed in OEIS A071519. None of them are prime.

Numbermatics representation of 26409

Today my diurnal age is 26409 and this number is a member of OEIS A071519, which is what drew my attention to the number's property:

If we allow zero and consider possible permutations of the digits from 0 to 9, there are 3,265,920 possibilities (9 x 9!) but only 87 are perfect squares and again, none are prime. All are divisible by 9. These 87 numbers form OEIS A156977:

A156977

Numbers \(n\) such that \(n^2\) contains every decimal digit exactly once.

The numbers are:

32043, 32286, 33144, 35172, 35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 40545, 42744, 43902, 44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55446, 55524, 55581, 55626, 56532, 57321, 58413, 58455, 58554, 59403, 60984, 61575, 61866, 62679, 62961, 63051, 63129, 65634, 65637, 66105, 66276, 67677, 68763, 68781, 69513, 71433, 72621, 75759, 76047, 76182, 77346, 78072, 78453, 80361, 80445, 81222, 81945, 83919, 84648, 85353, 85743, 85803, 86073, 87639, 88623, 89079, 89145, 89355, 89523, 90144, 90153, 90198, 91248, 91605, 92214, 94695, 95154, 96702, 97779, 98055, 98802, 99066

On July 9th 2018, I wrote about Pandigital Numbers Formed From the Product of a Number and its Reversal.