For positive $k$, define the function $H_k$ over the integer interval $[0\dots2^{2k})$ with $$H_k(n)\;=\;\left\lfloor{n^2\over2^k}\right\rfloor\bmod 2^{2k}$$ that is, keeping the middle $2k$ bits of the $4k$-bit square of $2k$-bit $n$. See Donald Knuth's TAOCP, chapter three (at start of volume 2) for references about use of middle square as a building block in pseudo-random number generators.
For large $k$, e.g. $64$ or $128$, can we exhibit $h\in[0,2^{2k})$ demonstrably reached by no $H(n)$?
If $n<2^{k/2}$, then $H_k(n)=0$. By a counting argument, it follows that several $h$ in $[0\dots2^{2k})$ are reached by no $H_k(n)$.
Numerical experiments show that such unreachable values are relatively dense just below $2^{2k}$ (more than they are just above $0$).
k (dec) Six highest unreachable values (hex)
1 3 1
2 F D B 7 5 3
3 3F 3B 39 36 35 31
4 FD FC FB F7 F5 F3
5 3FB 3F9 3F7 3F6 3F3 3F1
6 FFF FF8 FF7 FF3 FF2 FEF
7 3FFF 3FFD 3FFB 3FFA 3FF9 3FF6
8 FFFF FFFD FFFC FFF5 FFF4 FFF1
9 3FFFF 3FFFC 3FFFA 3FFF8 3FFF7 3FFF3
10 FFFFD FFFF9 FFFF8 FFFF6 FFFF5 FFFF1
11 3FFFF9 3FFFF7 3FFFF6 3FFFF3 3FFFEF 3FFFED
12 FFFFFD FFFFFB FFFFF7 FFFFF6 FFFFF4 FFFFF2
13 3FFFFEF 3FFFFEE 3FFFFED 3FFFFE8 3FFFFE5 3FFFFE3
14 FFFFFFD FFFFFFB FFFFFF5 FFFFFF4 FFFFFEF FFFFFEA
15 3FFFFFFF 3FFFFFFD 3FFFFFFA 3FFFFFF6 3FFFFFF4 3FFFFFF1
16 FFFFFFFD FFFFFFFC FFFFFFFA FFFFFFF9 FFFFFFF8 FFFFFFF7
17 3FFFFFFFD 3FFFFFFF8 3FFFFFFF7 3FFFFFFEF 3FFFFFFED 3FFFFFFEC
18 FFFFFFFFF FFFFFFFFB FFFFFFFFA FFFFFFFF7 FFFFFFFF4 FFFFFFFEC
Note: This is the unsolved portion of an old question on crypto-SE. A simplified mathoverflow question asking if $H_{64}(n)$ can reach $2^{128}-1$ was answered in the affirmative.
