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A109018
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Least n-almost prime number which gives a different n-almost prime number when digits are reversed.
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11
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13, 15, 117, 126, 270, 2576, 8820, 16560, 21168, 46848, 295245, 441600, 846720, 4078080, 80663040, 40590720, 2173236480, 4011724800, 21122906112, 40915058688, 274148425728, 63769149440, 2707602702336, 6167442456576
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OFFSET
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1,1
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COMMENTS
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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).
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).
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.
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).
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).
The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).
The Mathematica code for this was written by Ray Chandler who has coauthorship credit for this sequence.
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LINKS
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Eric Weisstein's World of Mathematics, Emirp.
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EXAMPLE
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a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.
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).
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.
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MATHEMATICA
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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 *)
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 *)
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CROSSREFS
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Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.
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KEYWORD
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base,nonn,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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