# [seqfan] Re: a somewhat silly problem

Don Reble djr at nk.ca
Sat Aug 22 04:48:12 CEST 2020

```> %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: