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Sep 28, 2017 at 8:27 comment added user114642 A distribution of the last digits of first 100 numbers for which sum of digits of 3^n is divisible by 7 shows some bias.
Sep 28, 2017 at 0:11 comment added Steven Stadnicki @SylvainJULIEN Even if one expects to have normality, note that the sum of $n$ uniformly distributed digits still has SD $\approx\sqrt{n}$, so much too large for approximations of the sort you're suggesting to be consistent to within the subunit ranges needed for it to do any good.
Sep 27, 2017 at 7:16 comment added Yaakov Baruch @WillSawin. Yes, for $n\le 20000$ the observed sdev (off the expected $4.5\log_{10}(3^n)$) is $1.00452$ times the value you gave, and the small local fluctuations within the interval show no trends or other remarkable features.
Sep 26, 2017 at 20:53 comment added Will Sawin @YaakovBaruch Is the jitter of size $\sqrt{8.25 \log_{10} (3^n)}$, as would be predicted by the random model if I calculated correctly?
Sep 26, 2017 at 19:51 comment added Gerhard Paseman You use gawk -M too? Gerhard "Prototyping Near To My Heart" Paseman, 2017.09.26.
Sep 26, 2017 at 19:39 comment added Yaakov Baruch Also, for $n \le 20000$ (working with simple "gawk -M" commands) I see for each of the 7 possibilities (mod $7$) runs of 4 or 5 consecutive equal values, as well as gaps of at least 50 in their occurrence.
Sep 26, 2017 at 19:22 comment added Yaakov Baruch As Gerhard suspects, for $2\le n\le 10000$, this is the unremarkable frequency of the values of the sum of digits (mod $7$): 0 1374, 1 1467, 2 1408, 3 1418, 4 1466, 5 1458, 6 1408.
Sep 26, 2017 at 19:19 comment added Sylvain JULIEN I mean that n should be large enough to make the normality hypothesis that Gerhard mentions hold. For rather small values of n, it is more convenient to use a computer.
Sep 26, 2017 at 19:12 comment added Yaakov Baruch @SylvainJULIEN: certainly not! A plot of sum_of_digits($3^n$) shows a lot of random jitter of increasing amplitude (but decreasing amplitude if one plots log(sum_of_digits) instead). I think this question is much like asking whether there is a shortcut to check if the sum of the first $n$ digits of, say, $\pi$ is a multiple of $7$ - but with a misleading number theory flavor that makes one think it possible.
Sep 26, 2017 at 18:56 comment added Sylvain JULIEN Quite interesting. Computing $ 4.5(\log_{10}3^{1000}-1) $ I get $ 2142.5456... $ so perhaps rounding the decimal log to the greatest even integer below it would do the job.
Sep 26, 2017 at 18:40 history answered Gerhard Paseman CC BY-SA 3.0