A119684 Ternary emirpimes.

112, 211, 1021, 1102, 1201, 2011, 2022, 2202, 10111, 11101, 11112, 12102, 12202, 12212, 20121, 20212, 20221, 21111, 21202, 21221, 100102, 100201, 101011, 101122, 101221, 102001, 102002, 102012, 102022, 102122, 102222, 110101, 110102, 110122, 110211, 111102, 111202, 112011, 112121, 112122, 112202

Offset: 1

Author

Jonathan Vos Post, Jun 08 2006

Keywords

• base
• easy
• nonn

Comments

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.

Apparently numbers with trailing zeros (reversed with leading zeros), like 1220 and 10020, are not included. - R. J. Mathar, Dec 22 2010

Examples

			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.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein, Jonathan Vos Post, et al., Emirpimes.
• Eric Weisstein, Vincenzo Origlio, et al., Ternary.

Crossrefs

Cf. A001358, A004086, A007089, A097393.

Programs

Maple

R:= NULL: count:= 0:
for m from 2 while count < 100 do
for j from 1 to 2 while count < 100 do
  n:= 3*m+j;
  if numtheory:-bigomega(n) <> 2 then next fi;
  L:= convert(n,base,3);
  r:= add(L[-i]*3^(i-1),i=1..nops(L));
  if r <> n and numtheory:-bigomega(r) = 2 then
     count:= count+1; R:= R, add(L[i]*10^(i-1),i=1..nops(L))
  fi
od od:
R; # Robert Israel, Jun 07 2020

Mathematica

(* 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 *)

Formula

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.

Extensions

More terms from Robert Israel, Jun 07 2020