A217379 - OEIS

%I #57 Mar 21 2021 13:05:36

%S 1,2,3,7,12,19,23,72,103,186,349,476,1955,42806,165541,191131,700419,

%T 874395,2702144,4408232,5397917,5442858,8751594,32886598,54013149,

%U 1274902129,1337169719,1419213312,2141194665,53879079616,100000000001

%N Numbers having non-pandigital power of record size (excludes multiples of 10).

%C With a search limit of 10^399, largest powers lacking a digit in decimal were compared with the record. Starting with a(2), the exponents are 168, 106, 61, 51, 44, 50, 39, 42, 39, 39, 41, 39, 30, 27, 27, 25, 27, 25, 25, 27, 27, 27, 26, 28, 24, 24, 25, 26, 24 and 24. a(3) = 3, a(19) = 2702144 and a(22) = 5442858 would have been excluded were the record based on number of digits rather than value; and, in fact, 3^106 is greater than 2^168 by < 0.4%. The largest power so far is 217 digits.  Prior to starting to test powers of 8-digit numbers, direct computations gave less than a 0.5% chance of finding a power > 249 digits for any number under 10^8.  The last several values are consistent with expectations.

%C The records claimed are unlikely to be proved--given the mysterious nature of digits prior to their actual computation, but the following provides some computational basis for them.

%C Direct computation on sizes of powers of numbers < 10^7 of 400-599 digits, assuming uniform distribution of digits, gives a pseudo-probability on the order of 10^(-10) for not all being pandigital; and, with the probabilities comparable to geometric series, the likelihood of even longer ones is even more remote.

%C The sequence is infinite because there are infinite non-pandigital powers. For example, for m in {2,3,...,19,22,24}, and any k>=m, the m-th power (1+10^k)^m is non-pandigital. - _Giovanni Resta_, Oct 26 2012

%H Jens Kruse Andersen, <a href="/A217379/a217379.txt">Powers and missing digits for n = 1..31</a>

%H Giovanni Resta, <a href="/A217379/a217379.c.txt">C program</a>

%e 2^168 = 374144419156711147060143317175368453031918731001856 lacks the digit 2; and all 2^p < 10^399, 168 < p, are pandigital.

%e 3^106 = 375710212613636260325580163599137907799836383538729 lacks the digit 4, and similarly. Since larger than 2^168, 3 is included. 4, as a power, is naturally excluded, and 5 and 6 do not give a record-size power (while 7 does).

%e As another example, 349^39 is a 100-digit number without the digit 0, and is the first power found in search to be non-pandigital and greater than 186^39.

%K nonn,base

%O 1,2

%A _James G. Merickel_, Oct 01 2012

%E a(25) added by _James G. Merickel_, Oct 21 2012

%E a(26)-a(31) added by _Giovanni Resta_, Oct 26 2012