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Search: id:A109019
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| A109019 |
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Numbers whose digit reversal is different and has the same number of prime factors (with multiplicity). |
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+0
2
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| 13, 15, 17, 26, 31, 37, 39, 49, 51, 58, 62, 71, 73, 79, 85, 93, 94, 97, 107, 113, 115, 117, 122, 123, 126, 129, 143, 147, 149, 155, 157, 158, 159, 165, 167, 169, 177, 178, 179, 183, 185, 187, 199, 203, 205, 221, 225, 226, 244, 246, 265, 270, 285, 286, 289, 290
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This sequence is the union of emirps, emirpimes, "emirp tsomla-3", "emirp tsomla-4", "emirp tsomla-5" and so forth. 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. He also has more values.
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LINKS
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Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.
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FORMULA
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{A006567} U {A097393} U (A109023} U {A109024} U ... U {A109131} U ...
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MATHEMATICA
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Select[Range[400], # != FromDigits[Reverse[IntegerDigits[ # ]]] && Sum[FactorInteger[ # ][[i, 2]], {i, 1, Length[FactorInteger[ # ]]}] == Sum[FactorInteger[ FromDigits[Reverse[IntegerDigits[ # ]]]][[i, 2]], {i, 1, Length[FactorInteger[FromDigits[Reverse[IntegerDigits[ # ]]]]]}] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jun 16 2007
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CROSSREFS
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Cf. A006567, A097393, A109018, A109023-A109131.
Sequence in context: A070603 A031060 A120129 this_sequence A068893 A079829 A096090
Adjacent sequences: A109016 A109017 A109018 this_sequence A109020 A109021 A109022
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KEYWORD
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nonn,base
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 16 2005
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EXTENSIONS
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More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jun 16 2007
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