A109018 - OEIS

%I #24 Jun 04 2024 12:23:16

%S 13,15,117,126,270,2576,8820,16560,21168,46848,295245,441600,846720,

%T 4078080,80663040,40590720,2173236480,4011724800,21122906112,

%U 40915058688,274148425728,63769149440,2707602702336,6167442456576,21586195906560,29798871072768,420127895977984,631722992467968

%N Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

%C An emirp ("prime" spelled backwards) is a prime whose (base 10) reversal is also prime, but which is not a palindromic prime. The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (A006567).

%C An emirpimes ("semiprime" spelled backwards) is a semiprime whose (base 10) reversal is a different semiprime. A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (A097393).

%C An "emirp tsomla-3" ("3-almost prime" spelled backwards) is the k=3 sequence of the series for which k=1 are emirps and k=2 are emirpimes, a list of these being A109023. The union of these for k=1 through k = 13 is A109019.

%C The primes correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (A000040). The 2-almost prime numbers correspond to semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... (A001358).

%C The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... (A014613).

%C The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).

%C The Mathematica code for this was written by _Ray Chandler_, who has coauthorship credit for this sequence.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Emirp.html">Emirp</a>.

%H Eric Weisstein and Jonathan Vos Post, <a href="http://mathworld.wolfram.com/Emirpimes.html">Emirpimes</a>.

%e a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.

%e a(2) = 15 because 15 is the smallest emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, since 15 reversed is 51 and both 15 and 51 are semiprimes (i.e. 15 = 3 * 5 and 51 = 3 * 17).

%e a(3) = 117 because 117 is the smallest "emirp tsomla-3" ("3-almost prime" spelled backwards) since 117 reversed is 711 and 117 = 3^2 * 13 and 711 = 3^2 * 79.

%t kAlmost[n_] := Plus @@ Last /@ FactorInteger@n; fQ[n_] := Block[{id = IntegerDigits@n, k = kAlmost@n}, If[id != Reverse@id && k == kAlmost@FromDigits@Reverse@id, k, -1]]; t = Table[0, {20}]; Do[ a = fQ@n; If[a < 20 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 10, 150000000}] (* _Robert G. Wilson v_, Jan 06 2008 *)

%t Table[Select[Range[41*10^5],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #] ==n&][[1]],{n,14}] (* The program generates the first 14 terms of the sequence. *) (* _Harvey P. Dale_, Oct 15 2023 *)

%Y Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

%K base,nonn,less

%O 1,1

%A _Jonathan Vos Post_, Jun 16 2005

%E a(14)-a(16) from _Robert G. Wilson v_, Jan 06 2008

%E a(17)-a(24) from _Donovan Johnson_, Nov 17 2008

%E a(25)-a(28) from _Michael S. Branicky_, Jun 04 2024