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A158594
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Numbers which yield a prime whenever a 3 is prefixed, appended or inserted
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3
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1, 7, 11, 17, 31, 37, 73, 121, 271, 331, 343, 359, 361, 373, 533, 637, 673, 733, 793, 889, 943, 1033, 1183, 2297, 3013, 3119, 3223, 3353, 3403, 3461, 3757, 3827, 3893, 3923, 4313, 4543, 4963, 5323, 5381, 5419, 6073, 6353, 8653, 9103, 9887, 10423, 14257
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 1) It is conjectured that sequences of this type are infinite; also that an infinite number of primes is included
2) Necessarily a(n) has end digit 1,3,7 or 9
3) Sum of digits of a(n) has form 3k-1 or 3k+1
4) Sequence is part of A068674 a(n) n=1,...,30: 14 first primes: 7, 11, 17, 31, 37, 73, 271, 331, 359, 373, 673, 733, 2297, 3461
5) Note the "world record" 2297: smallest prime which yields five other primes 32297,23297,22397,22937,22937
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REFERENCES
| Marcus Du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins. 2004
Bryan Bunch, Kingdom of Infinite Number: A Field Guide, W.H. Freeman & Company, 2001
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EXAMPLE
| 1) 109 is no member: 3109,1039,1093 are primes, but 1309 = 7 x 11 x 17
2) 121: 3121 (3 prefixed), 1213 (3 appended), 1321 and 1231 (3 inserted) are primes
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MAPLE
| Lton := proc(L) local i ; add(op(i, L)*10^(i-1), i=1..nops(L) ) ; end: isA158594 := proc(n) local dgs, i, p; dgs := convert(n, base, 10) ; p := [3, op(dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; p := [op(dgs), 3] ; if not isprime(Lton(p)) then RETURN(false) ; fi; for i from 1 to nops(dgs)-1 do p := [op(1..i, dgs), 3, op(i+1..nops(dgs), dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; od: RETURN(true) ; end: for n from 1 to 25000 do if isA158594(n) then printf("%d, ", n) ; fi; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 26 2009]
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CROSSREFS
| A068674 Numbers which yields primes when a 3 is prefixed or appended
A068679 Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end)
A158232 Numbers which yield primes when "13" is prefixed or appended
Sequence in context: A019418 A068674 A156112 * A107642 A134572 A079651
Adjacent sequences: A158591 A158592 A158593 * A158595 A158596 A158597
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KEYWORD
| nonn,base
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AUTHOR
| Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 22 2009
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EXTENSIONS
| Corrected and extended by Chris K. Caldwell (caldwell(AT)utm.edu) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 26 2009
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