[seqfan] Re: a somewhat silly problem

[Jeffrey Shallit]

I specified in my initial posting on seqfan that I wasn't allowing 
leading zeroes.

I don't understand why you disallow 1028^2.

On 2020-08-21 10:48 p.m., Don Reble via SeqFan wrote:
>> %I A337252
>> %S 8,10,12,14,20,26,28,30,32,34,36,38,40,...
>> %N Digits of 2^n can be rearranged to form t^2, for t not a power of 2.
>> %e 2^56, 142847044^2
>> %e 2^58, 318068365^2
> 
>     Define this one carefully: 2^22 = 4194304 -> 0413449 = 643^2.
> 
>     Here are yet more successes. (My program orders the search-space
>     differently.)
> 
> 60 2338710110
> 62 2912670110
> 64 7844230210
> 66 6105463102
> 68 24123141101
> 70 87430611010
> 72 70475261111
> 74 280158623201
> 76 183175311011
> 78 611263401110
> 80 2485552010110
> 82 2120742201110
> 84 4346880101101
> 86 4433461101011
> 88 31425860110100
> 90 62277440011010
> 92 44440611620110
> 94 241792452111100
> 96 222703241001101
> 98 870331350210100
> 100 1955701101011000
> 102 1628483001101000
> 104 6054623000110100
> 106 4466731320201100
> 108 29057819711010110
> 110 53628010101011000
> 112 99471547000111010
> 114 182717481101011010
> 116 279208944301010110
> 118 597042351010101100
> 120 2549117011100110100
> 122 1982810111210001101
> 124 3958021601110001101
> 126 6037703202020101100
> 128 19908408612100011010
> 130 99173527101110001101
> 132 74795637011101011010
> 134 199391022011101011010
> 136 139885442010101010110
> 138 765091430120202011000
> 140 1839416746111100011010
> 
>     It looks rather dense. Perhaps one should use the complementary
>     sequence, and use 4^n instead of 2^n:
> 
> %I A337252
> %S 0,1,2,3,8,9,11,12
> %N The digits of 4^n cannot be rearranged to form the digits of t^2,
>    for t not a power of 2,
> %C 2^odd cannot be rearranged to a square number: odd powers of 2 are
>    congruent to 2,5,8 mod 9; squares are congruent to 0,1,4,7 mod 9;
>    and rearranging preserves the mod-9 value.
> %C a(9) > 70; but are there any more terms?
> %e 10 is not here, because 4^10 = 1048576 -> 1056784 = 1028^2.
> %e 11 is here, even though 4^11 = 4194304 -> 0413449 = 643^2:
>    leading zeroes aren't allowed.
>