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[parent] values of $4^n$ and $\displaystyle \prod_{i = 1}^{\pi(n)} p_i$ for $0 < n < 26$ (Example)

The following table is in support of the statement that

$\displaystyle 4^x > \prod_{i = 1}^{\pi(x)} p_i$
is always true for any nonnegative $ x$.
$ n$ $ 4^n$ $ \displaystyle \prod_{i = 1}^{\pi(n)} p_i$ $ \displaystyle 4^n - \prod_{i = 1}^{\pi(n)} p_i$
1 4 1 3
2 16 2 14
3 64 6 58
4 256 6 250
5 1024 30 994
6 4096 30 4066
7 16384 210 16174
8 65536 210 65326
9 262144 210 261934
10 1048576 210 1048366
11 4194304 2310 4191994
12 16777216 2310 16774906
13 67108864 30030 67078834
14 268435456 30030 268405426
15 1073741824 30030 1073711794
16 4294967296 30030 4294937266
17 17179869184 510510 17179358674
18 68719476736 510510 68718966226
19 274877906944 9699690 274868207254
20 1099511627776 9699690 1099501928086
21 4398046511104 9699690 4398036811414
22 17592186044416 9699690 17592176344726
23 70368744177664 223092870 70368521084794
24 281474976710656 223092870 281474753617786
25 1125899906842624 223092870 1125899683749754



"values of $4^n$ and $\displaystyle \prod_{i = 1}^{\pi(n)} p_i$ for $0 < n < 26$" is owned by PrimeFan.
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This is version 2 of values of $4^n$ and $\displaystyle \prod_{i = 1}^{\pi(n)} p_i$ for $0 < n < 26$, born on 2007-05-13, modified 2007-05-16.
Object id is 9374, canonical name is ValuesOf4nAndProd_i1pinP_iFor0N26.
Accessed 387 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
 11N05 (Number theory :: Multiplicative number theory :: Distribution of primes)
Pending Errata and Addenda
None.
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