%I #15 Feb 16 2025 08:33:01
%S 112,211,1021,1102,1201,2011,2022,2202,10111,11101,11112,12102,12202,
%T 12212,20121,20212,20221,21111,21202,21221,100102,100201,101011,
%U 101122,101221,102001,102002,102012,102022,102122,102222,110101,110102,110122,110211,111102,111202,112011,112121,112122,112202
%N Ternary emirpimes.
%C These are semiprimes when read as base 3 numbers and their reversals are different semiprimes when read as base 3 numbers. Base 10 these are: 14, 22, 34, 38, 46, 58, 62, 74, 94, 118, 122, 146, 155, 158, 178, 185, 187, ... See: A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes. See: A004086 Read n backwards (referred to as R(n) in many sequences). See: A007089 Numbers in base 3.
%C Apparently numbers with trailing zeros (reversed with leading zeros), like 1220 and 10020, are not included. - R. J. Mathar, Dec 22 2010
%H Robert Israel, <a href="/A119684/b119684.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein, Jonathan Vos Post, et al., <a href="https://mathworld.wolfram.com/Emirpimes.html">Emirpimes</a>.
%H Eric Weisstein, Vincenzo Origlio, et al., <a href="https://mathworld.wolfram.com/Ternary.html">Ternary.</a>
%F a(n) = A007089(i) for some i in A001358 and R(a(n)) = A007089(j) for some j =/= i in A001358. a(n) = A007089(i) for some i in A001358 and A004086(a(n)) = A007089(j) for some j =/= i in A001358.
%e a(1) = 112 because 112 (base 3) = 14 (base 10) is semiprime and R(112) = 211, where 211 (base 3) = 22 (base 10) is a different semiprime.
%e a(13) = 12202 because 12202 (base 3) = 155 (base 10) is semiprime and R(12202) = 20221, where 20221 (base 3) = 187 (base 10) is a different semiprime.
%p R:= NULL: count:= 0:
%p for m from 2 while count < 100 do
%p for j from 1 to 2 while count < 100 do
%p n:= 3*m+j;
%p if numtheory:-bigomega(n) <> 2 then next fi;
%p L:= convert(n,base,3);
%p r:= add(L[-i]*3^(i-1),i=1..nops(L));
%p if r <> n and numtheory:-bigomega(r) = 2 then
%p count:= count+1; R:= R, add(L[i]*10^(i-1),i=1..nops(L))
%p fi
%p od od:
%p R; # _Robert Israel_, Jun 07 2020
%t (* First run the program for A105999 *) SemiPrimeQ[n_Integer] := TrueQ[SemiPrimePi[n] > SemiPrimePi[n - 1]]; BaseForm[Select[Table[SemiPrime[n], {n, 100}], GCD[#, 3] == 1 && # != FromDigits[Reverse[IntegerDigits[#, 3]], 3] && SemiPrimeQ[FromDigits[Reverse[IntegerDigits[#, 3]], 3]] &], 3] (* From _Alonso del Arte_, Dec 22 2010 *)
%Y Cf. A001358, A004086, A007089, A097393.
%K base,easy,nonn,changed
%O 1,1
%A _Jonathan Vos Post_, Jun 08 2006
%E More terms from _Robert Israel_, Jun 07 2020