Python 3, 36 32 bytes lambda x:{*str(x)}=={*str(x**3)} I think this only works in Python 3.5 and later. Four bytes have gone, thanks to Copper.

Python 3, 54 50 bytes lambda n:sum(bin(i)[2:]in bin(n)for i in range(n)) Thanks to Rod and Jonathan Allan for saving four bytes.

Brainfuck, 140 bytes >>>++++++++[>++++++>++++++++++++<<-]++++++++++>->----<<<<<+++[>++++[>++++++[>>.....>.....<<<-]>.<<-]++++[...

Python 3, 150 bytes from itertools import*;lambda x:([i for i in range(10)if(*' OTTFFSSENNWHOIIEIIEORUVXVGN ERE EHE E NT '[i::9],)in[*combinations(x+' ',5)]]+) combinations returns all the ...

Python 3, 65 bytes lambda x:len(x)<18and x[:len(x)//2+1]in'0123456789'and x[::-1]==x I'm not entirely sure, but I think this works.

Python 2, 49 44 42 bytes f=lambda x:'3'[:~x%3]in~xand f(x+1)or-~x The other Python entry beats this (edit: not any more :-D), but I posted it because I rather like its recursive approach. Thanks to ...

Python 3, 28 bytes lambda x:min(hex(x)[1:])>'@'

Python 3, 57 bytes k=int(input());print(all(k%j for j in range(2,k))and k>1) I think it's pretty self-explanatory.

Python, 228 bytes from itertools import*;s,C=lambda x,y:sorted(sum([[i+j for j in y]for i in x],[])),combinations_with_replacement;f=lambda k,l,m:([[*a]for a in C([*C([*range(1,s(l,m)[-1])],k)],2)if(s(...

Python 3, 66 bytes r=range(16) for i in r:print(*[('0'+hex(j*i)[2:])[-2:]for j in r])

Python 3, 48 46 bytes f=lambda x,i=1:f(x,i+1)if(i+3)*i+2<x/i*6else i

Python 3, 128 bytes 0-indexed; takes in strings as parameters. -6 bytes thanks to Jonathan Frech. from sympy.ntheory import* def f(n,m):a=[sum(factorint(int(n[:i]+m+n[i:])).values())for i in range(len(...

Python 3, 68 bytes def f(a,b): c='|'+'-'*(a-2)+'|\n';print(c+c.replace(*'- ')*(b-2)+c)

Python 3, 48 bytes lambda c,k:''.join([c[i:-1:k]for i in range(k)])